ESTIMATED AND ANALYSIS OF THE RELATIONSHIP BETWEEN THE ENDOGENOUS AND EXOGENOUS VARIABLES USING FUZZY SEMI-PARAMETRIC SAMPLE SELECTION MODEL

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1 Amercan Journal of Appled Scences (9: , 204 ISSN: L. MuhamadSafh et al., Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton (CC-BY 3.0 lcense do:0.3844/ajas Publshed Onlne (9 204 ( ESTIMATED AND ANALYSIS OF THE RELATIONSHIP BETWEEN THE ENDOGENOUS AND EXOGENOUS VARIABLES USING FUZZY SEMI-PARAMETRIC SAMPLE SELECTION MODEL L. MuhamadSafh, 2 A.A. Kaml and 3 M.T. Abu Osman School of Informatcs and Appled Mathematcs, Insttute of Marne Botechnology, Unversty Malaysa Terengganu, 2030 Kuala Terengganu, Terengganu, Malaysa 2 School of Dstance Educaton, Unversty Sans Malaysa, 800 USM Penang, Malaysa 3 Department of Computer Scence, Kullayyah Informaton and Communcaton Technology, Internatonal Islamc Unversty Malaysa, P.O.Box 0, Kuala Lumpur, Malaysa Receved ; Revsed ; Accepted ABSTRACT An mportant progress wthn the last decade n the development of the selectvty model approach to overcome the nconsstent results f the dstrbutonal assumptons of the errors terms are made ths problem s through the use of sem-parametrc method. However, the uncertantes and ambgutes exst n the models, partcularly the relatonshp between the endogenous and exogenous varables. A new framework of the relatonshp between the endogenous and exogenous varables of sem-parametrc sample selecton model usng the concept of fuzzy modellng s ntroduced. Through ths approach, a flexble fuzzy concept hybrd wth the sem-parametrc sample selecton models known as Fuzzy Sem-Parametrc Sample Selecton Model (FSPSSM. The elements of vagueness and uncertanty n the models are represented n the model constructon, as a way of ncreasng the avalable nformaton to produce a more accurate model. Ths led to the development of the convergence theorem presented n the form of trangular fuzzy numbers to be used n the model. Besdes that, proofs of the theorems are presented. An algorthm usng the concept of fuzzy modellng s developed. The effectveness of the estmators for ths model s nvestgated. Monte Carlo smulaton revealed that consstency depends on bandwdth parameter. When bandwdth parameters, c are ncreased from 0., 0.5, 0.75 and as the numbers of N ncreased (from 00 to 200 and ncreased to 500, the values of mean approaches (closed to the real parameter. Through the bandwdth parameter also reveals that the estmated parameter s effcent,.e., the S.D, MSE and RMSE values become smaller as N ncreased. In partcular, the estmated parameter becomes consstent and effcent as the bandwdth parameters approaches to nfnty, c as the number of observatons, n tend to nfnty, n. Keywords: Selectvty Model, Sem-Parametrc, Fuzzy Concept, Bandwdth, Monte Carlo. INTRODUCTION The sample selecton model or the selectvty model ntroduced by Heckman (979 s one of the most successful regresson models f appled together wth other models. Ths model s a combnaton of the probt and on the parametrc approach. However, the standard approach of estmatng sample selecton model shows nconsstent results f the dstrbutonal assumptons of the errors terms are made. Hence, an mportant progress wthn the last decade n the development of an alternatve approach to overcome ths problem s through the use of regresson models. The earler studes on ths model focused sem-parametrc method (Andrews (99; Cosslett, 990; Correondng Author: L. MuhamadSafh, School of Informatcs and Appled Mathematcs, Insttute of Marne Botechnology, Unversty Malaysa Terengganu, 2030 Kuala Terengganu, Terengganu, Malaysa Scence Publcatons 542

2 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 Gerfn, 996; Ichmura and Lee, 99; Khan and Powell, 200; Klen and Spady, 993; Lee and Vella, 2006; Martns, 200; Powell, 987; Powell et al., 989. Although sem-parametrc method of selectvty model s establshed, there stll exst a basc problem of ntrnsc features, such as uncertanty and ambguty partcularly n the relatonshp between the endogenous and exogenous varables. Therefore, t wll dsrupt the ablty and effectveness of the model proceeded to gve the estmated value that can explan the actual stuaton of a phenomenon. These are questons and problems that have yet to be explored and the man pllar of ths study. A new framework of the relatonshp between the endogenous and exogenous varables of sem-parametrc sample selecton model usng the concept of fuzzy modellng by Zadeh (965 s ntroduced. Through ths approach, a flexble fuzzy concept hybrd wth the sem-parametrc sample selecton models known as Fuzzy Sem-Parametrc Sample Selecton Model (FSPSSM. Hence, an alternatve way to deal wth ths uncertanty and ambguty s to use fuzzy concepts ntroduced by Zadeh (965. The purpose of ths chapter s twofold; frstly, to provde a better understandng of the magntude of consstency as well as effcency, when the new modelng of FSPSSM s mplemented under normalty assumpton. It s then extended to verfy the nconsstency of the model when t does not follow the assumpton of the normal dstrbuton. Secondly, s to provde the magntude of the consstency under FSPSSM. For ths purpose, the bandwdth parameter of Powell (987 model s used. To acheve these ams, Monte Carlo smulatons usng R language programmng by Safh (203 and as well as the estmator ntroduced by (Powell et al., 989; Powell, 987 whch are hybrd wth fuzzy concept s developed. 2. MATRIALS AND METHODS 2.. The Sem-parametrc Sample Selecton Model (SPSSM The sem-parametrc sample selecton model s a hybrd between the two sdes of the sem-parametrc approach,.e., t combnes some advantages of both fully parametrc and the completely nonparametrc approaches. The frst model,.e., partcpaton equaton s estmated by the parametrc method, whle the outcome equaton s estmated by the nonparametrc method. For nstance, (Newey et al., 990; Martns, 200 used a two-step semparametrc approach of model (. The Sem-Parametrc Sample Selecton Model (SPSSM can be wrtten as: z = w γ + ε * = + > d = 0 otherwse z = z d ; =,..., N * f d x β u 0 * where, d and z ( are dependent varables x and w are vectors of remanng exogenous varablesγandβare unknown parameter vectors ε and u are error terms. It generalses the Heckman s two-step procedure,.e., n the frst step, the partcpaton equaton s estmated semparametrcally usng the estmator proposed by Klen and Spady (993. The results from ths frst step are used to construct a nonparametrc correcton term for selectvty of wage equaton n the second step. The dfference between parametrc and sem-parametrc approaches comes n the form of weaker assumpton of the error term. The two-step estmaton procedure refers to the estmaton of the partcpant and the outcome equatons as menton n Lola et al. (2009. Consder the bnary selecton model and proceeds by ecfyng the parametrc part of the model. Order the N observatons such that the frst,...,n observatons represent partcpants wth d = and y observed. The remanng observatons were the nonpartcpants wth d = 0 and y unobserved. In ths study, the sem-parametrc method n Equaton s consdered. As wth the Safh (203 paper of FPSSM, ths method nvolves two steps. In the frst step, the parameter, β n the partcpaton equaton s estmated usng Densty Weghted Average Dervatve Estmator (DWADE and n the second step, Powell (987 estmator n the outcome equaton s used to estmate the parameter, γ. The DWADE whch was proposed by Powell et al. (989 s used to estmate parameter βn the frst step of Equaton. Ths estmator s based onsample analogues of the product moment representaton of the average dervatons and s constructed usng nonparametrckernel estmators of the densty of the regressors. However, a practcal nterest of weghted average dervatves s that they are proportonal to coeffcents vector βn the ndex functon. Powell estmatorproposed by Powell (987 s used to estmate parameter γn the second step of Equaton. Powell (987 consdered asem-parametrc selecton model that combnes the two-equaton structure wth the followng weak dstrbuton alas sumpton about the jont dstrbuton of the error terms wth the form: ( f ( ε, u w = f ε, u wγ (2 Scence Publcatons 543

3 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 It s assumed that the jont densty of ε, u (condtonal on w s smooth but wth unknown functon f(.. Hence t depends on w only through the lnear model,.e., wγ. Based on these assumptons, the regresson functon for the observed outcome z takes the followng form: * * ( = (, > 0 E z x E z w d = wγ + E ( u w, x β > ε = wγ + λ β ( x where, λ(. s an known smooth functon. Ideally, gven two observatons and j wth w w j and the condton w γ = w γ the unknown functon λ(. canbe dfferentated of j by subtractng the regresson functons for and j: E z w = w E z w = w = * * ( ( j j w w γ λ x β λ x β ( = j + ( ( j = ( w w γ j Ths s the basc dea underlyng the estmator of γ as proposed by Powell (987: N N n ˆ γ ˆ powell = ϖ jn ( w wj ( w wj ` 2 = j = + N N n. ϖˆ jn( w wj ( z z j 2 = j = + These weghts of ϖˆ j are calculated, ˆ γ can be estmated by a weghted least-squares estmator, where ˆ x ˆ β x jβ ϖˆ jn = k wth symmetrc kernel functon c c k(., bandwdth c. and the estmate parameter ˆβ has already obtaned prevously, as an estmate of β. Under Equaton 2, we obtan a sngle ndex model for the decson equaton n place of the probt model (probt step n the parametrc case Equaton 4: ( ( 0 ( (3 P d d > x = = g x β (4 where, g(. s unknown but a smooth functon. Estmators for ˆβ n ths model have been dscussed n secton 2.5, as the frst step of the sem-parametrc procedure. Gven ˆ γ, the second step of the sem-parametrc procedure conssts of estmatng γ usng Equaton 3. Powell (987 proved that the ˆpowell γ estmator n Equaton 3 s n -consstent and asymptotcally normal under an approprate chosen Kernel (or bandwdth c. Ths result provded a n -consstent and asymptotcally normal dstrbuton as Equaton 5: d n( ˆ γ γ N (0, V (5 powell powell d where, denotes convergence n dstrbuton. The Powell procedure takes the data as nput from the outcome equaton (x and y, where may not contan a vector of ones. The frst-step, ndex x ˆ β s estmated whch nvolved the vector d and bandwdth vector, c. Both d and c are DWADE whch are multply by an..d. random sample and the k threshold parameter, reectvely. The frst element of c s used to estmate the ntercept coeffcent. The bandwdth c from the second element s used for estmatng the slope coeffcents Fuzzy Modellng In ths study, we used the fuzzy set defnton that s related to the exstng fuzzy set theory ntroduced by Zadeh (965. The Defnton of fuzzy numbers s followed from Yen et al. (999. The defnton s as follows: Defnton The fuzzy functon s defned by f : X A% Y% ; Y% = f ( x, A% where: A X;X s a cr set A % s a fuzzy set Y % s the codoman of x assocated wththe fuzzy set A % Some propertes of fuzzy set where A F (R s called a fuzzy number f: There exst x Rsuch that µ A (x = α A = [x, µ αa (x a] s a closed nterval for every α (0,] where,r s the set of real numbers The membershp functon for Trangular Fuzzy Number (TFN, µ A (x: R [0,] s descrbed as below Equaton 6: ( x f x [ I, m] ( m f x = m µ A( x = ( u x f c [ m, u] ( u m 0 otherwse (6 Scence Publcatons 544

4 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 where, m u, x s a value of real number I and u, the lower and upper bound of the support of A, reectvely. Then the TFN s denoted by (l, m,n. The support of s the set elements {x R l<x<u}. A nonfuzzy number by conventon occurs when l = m = u. Defnton 2 Let X be a ace of pont and x X, x D X s.t. µ D : X [0,]. Then D % = {(x, µ D(x} s called a fuzzy data. The process for gettng fuzzy data s llustrated n Fg.. In ths fgure, x s orgnal data (Fg. a whch nvolves uncertanty. Hence, t s called cr uncertanty data (Fg. b whch s assgned the value of or 0. In order to get a fuzzy data, the process of fuzzfcaton (Fg. c wth the membershp functon between (0,] and defuzzfcaton (Fg. d wll be mplemented. The structure of fuzzy data, ecfcally the process of fuzzfcaton s depcted n Fg. 2 wthn the α -cut. The lower and upper bound of each observaton follows the trangular membershp functons β LI ( ϖ, α = ( α and β 2 UI ( ϖ, α = ( α and become lower and upper bound reectvely. Where: And: Theorem LI ( ϖ = LI ( ϖ% + α ( ϖ LI ( ϖ% ( α ( α ( UI ( ϖ = UI ( ϖ% + α ϖ UI ( ϖ% Let the fuzzy data be defned by TFN, then the coeffcent values of the exogenous varables of the partcpaton and wage equatons for fuzzy data converge to the coeffcent values of exogenous varables of the partcpaton and wage equatons for cr data, reectvely, whenever the value of α-cut tend to. Fg.. Step by step process of gettng fuzzy data (a Orgnal data (b Cr uncertanty data (c Fuzzfcaton (d Defuzzfcaton Scence Publcatons Fg. 2. Membershp functon and t s -cut 545

5 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 Proof In order to get the cr value, the centrod method s used. Then, the fuzzy number for all observatons of ϖ s gven as: (( ( ϖ ϖ ( ϖ W = LI + + UI c 3 If α, then the values of µ ( x =, where the αa lower bound and upper bound for each observaton s based on Equaton 2. Applyng the α-cut nto the trangular membershp functon, the fuzzy number that s obtaned depends on the gven value of the α-cut over the range 0 and and s as follows: LI ( ϖ + α ( ϖ LI( ϖ + ϖ + UI( ϖ UI ( ϖ W = c ( α 3 = ( LI ( ϖ ( α + ϖ + UI ( ϖ ( α 3 used whch nvolved 3 stage.e., ( fuzzfcaton, (2 fuzzy envronment and (3 defuzzfcaton. In the frst stage, the elements of real-valued nput varables or cr uncertanty values are converted nto fuzzy data usng a partcular value of membershp functon. A trangular fuzzy number wth α-cut method s used for all observatons. In ths study, the same α-cuts method as n Equaton 7 s consdered. Hence, lower and upper bounds for each observaton ( w *, x, z,, u obtaned whch s defned reectvely as: w = ( w, w, w, x = ( x x, x l m u l, m u * l m u l m u ε s z = ( z, z, z, ε = ( ε, ε, ε and u = ( u, u, u l m u Accordng to Equaton 2, ther reectve membershp functons are defned as: When α approaches, then: Ll ( ϖ ϖ and Ul ( ϖ ϖ Further, we obtaned: ( α (a c ( a ( % W ϖ + ϖ + ϖ = ϖ l u 3 µ A k =,2,3,4,5 ( x α K l f x [ α, α ] K ( a α l Kl f x = α ( x = ( α x Ku f x [ α, α [ ( α a Ku Ku 0 otherwse (8 W Hence: c( α Wc ( α ϖ (7 Equaton 7 stated that when α approaches, then approaches cr, ϖ. In general, any observaton of the real fuzzy data s cr for all observatons such that x and z, X x and Z z reectvely, as αtends to c( α c ( α. Ths mples that the fuzzy data values of the partcpaton and structural equatons converge to the values of the partcpaton and structural equatons for cr data, reectvely whenever the value of α-cuts tend to Development of Fuzzy Sem-Parametrc Sample Selecton Model In order to formulate a fuzzy SPSSM, the SPSSM n Equaton s reconsdered. Towards the development of FSPSSM, the same procedure n Lola et al. (2009 s Based on Equaton 8, 5 types of membershp functon µ, can be generated by A takng on the Ak values of K =, 2, 3, 4 and 5, where K =, 2, 3, 4 and 5 * represents w, x, z, ε and u reectvely. For the second step, the α-cuts method s proceeded for all the exogenous varables and error terms. Whle for the thrd stage, the fuzzy values are converted to output of the cr value as n the followng formula Equaton 9: A K =,.., k ( α, α, α k Kl K u = (9 K = where, α and α Ku represents lower and upper bounds, K l reectvely. Agan, 5 types of defuzzfed can be generated n ths stage where AK takes values of K =, 2, 3, 4 and 5. The values of, 2, 3, 4 and 5 representng K Scence Publcatons 546

6 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 w, x, z*, ε and u reectvely. Hence, the FSPSSM s of the form as Equaton 0: Z% = w% γ + % ε * f d% = x% + u% > d = 0 otherwse z% = z% d ; =,..., N * * β 0 * (0 * The terms w %, x %, z %, ε and u% are fuzzy numbers wth the membershp functons µ, µ, µ *, µ and µ reectvely. w x z ε u 2.4. The Monte Carlo Smulaton Consstency and Effcency of FSPSSM To obtan a consstent estmator of FSPSSM n Equaton, the error terms s assumed to follow a normal dstrbuton. The hybrd of the model proposed by Nawata (994 wth fuzzy concept s consdered. Then, the Monte Carlo smulaton technque (Kabaday, 2004; Rana et al., 2008; 2009; Wtchakul el al., 2008 s used to llustrate the developed model. Adversely, the estmators are nconsstent f the error terms does not satsfy normal dstrbuton (Chamberlan, 986; Robnson, 988; Powell et al., 989; Cosslett, 990; Ichmura and Lee, 99; Newey et al., 990; Vella, 992; Ichmura, 993; Schafgans, 996; Markus, 998. For the development of SPSSM, one of the elements used to measure the consstency or effcency of the parameter s through the usage of bandwdth parameter, c(for nstance, Chamberlan, 986; Powell et al., 989; Andrews, 99; Cosslett, 990; Ahn and Powell, 993; Klen and Spady, 993; Schafgans, 996; Das et al., 2003; Bellemare et al., Accordng to Hardle (990, the bandwdth parameter s a scalar argument to the kernel functon that determnes what range of the nearby data ponts wll be heavly weghted n makng an estmate. The choce of bandwdth represents a trade-off between bas (whch s ntrnsc to a kernel estmator and whch ncreases wth bandwdth and varance of the estmates from the data (whch decreases wth bandwdth. An estmator s effcent f the RMSE values become smaller as the bandwdth parameter, c values ncreases as the number of N ncreased The Monte Carlo Smulaton of Fuzzy Sem- Parametrc Sample Selecton Model As mentoned earler to acheve the second am of ths study.e., consstency under FSPSSM, the Monte Carlo smulaton developed by Nawata (994 wth α-cuts of 0.2, 0.4, 0.6 and 0.8 are also consdered. In ths secton, the effectveness of the proposed model s focused on the usage of bandwdth parameter, c. Therefore, the form of FSPSSM wth DWADE and Powell estmators are hybrd wth Nawata (994 can be rewrtten accordng to Equaton as follows: y% = b + b w% + % ε 0 d = ( α + α x% + u% > 0, =,2,3,.., N 0 ( The values of w% and x% n the partcpatonand selecton equaton are ndependently, dentcally dstrbuted (..d random varablehavng unform dstrbuton wth the meansvalue of 0 and varance of 20. ρs thecorrelaton coeffcent between fuzzyexogenous varables, ( w% and x%. In Equaton, the exogenous varables, x% and w% whch nvolve bandwdth parameter, care followed from DWADE and Powell procedures, reectvely. The fuzzy error terms of ε% and u% are..d normal random varables. For ε% theestmated parameters of the means s zero andvarance s. Meanwhle, for u% theestmated parameter of the mean s zero andstandard devaton s 0.ϑ 0 s the correlatoncoeffcent between fuzzy error terms, ( ε% and u %. The performance of the FSPSSM under consstency wth bandwdth parameters,c, due to tables reducton, we show only thevalues of ρand 0 ϑ 0 s 0,.e., representng parameter of no correlaton between w% and x%, ε% and u%, reectvely. As per secton 5.3, the value of ϑ belongs to set [-,] and 0 s chosen as the mddle value of tsset. The sample sze of N = 00, and the bandwdth parameter, c = 0., 0.5, 0.75, are consdered. The bandwdths parameters, cgoverns the degree of smoothness mposedon the estmated functon f (. as n secton 2.5, wth large values of c correondng to asmoother functon estmate (Newey et al., 990. For all cases, the number ofreplcatons s,000. The true parameters value of γ s. Scence Publcatons 547

7 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , RESULTS 3.. The Monte Carlo Smulaton Result: The Fuzzy Sem-Parametrc Sample Selecton Model The results of Monte Carlo smulaton of FSPSSM wth N = 00, 200 and 500 are presented n Table to 3, reectvely. The frst and second column are the α-cuts.e., 0.2, 0.4, 0.6, 0.8 and bandwdth parameters, c.e., 0., 0.5, 0.75,, reectvely. The rest of the columns represent the mean, the Standard Devaton (S.D, the Mean Square Error (MSE and the Root Mean Square Error (RMSE, reectvely. To study the consstency under FSPSSM, we only reported the Powell estmator. Ths s due to the DWADE estmator s estmated and used nsde the Powell estmator. The consstency results of FSPSSM are obtaned n Table. The table shows that when N = 00, α-cuts = {0.2, 0.4, 0.6 and 0.8} and bandwdth parameters, c = 0., the means values of FSPSSM are , and When bandwdth parameters ncreases to 0.5, 0.75 and wth α-cuts = {0.4, 0.6 and 0.8}, the values of mean approaches (closed to the true parameter of γ,.e.,.06847,.06839, , ; ,.06705,.06709, and , , , , reectvely. These ndcated that the parameter estmates are consstent under bandwdth condton. A part of the consstency usng bandwdth parameters, t s used also to performed an effcent of the estmated parameters. Ths s reported n Table and 3 by the S.D, MSE and RMSE values. For nstance, Table shows that when N s 00, α-cuts s 0.2 and bandwdth parameter, c s 0., the S.D, MSE and RMSE values reectvely as follows: , and When the bandwdth parameters, c, ncrease to 0.5, 0.75 and, the S.D, MSE and RMSE values decreased, reectvely as , , ; , , and , , The same results are also shown for the α-cuts of 0.4, 0.6 and 0.8,.e., the S.D, MSE and RMSE values are decreased as bandwdth parameters, c ncreased (from 0., to 0.5, 0.75 and. For nstance, when α-cuts = 0.8 and bandwdth parameter, c = 0., the S.D, MSE and RMSE values are , and These values are decreased as ncreases of bandwdth parameters, c (0.5, 0.75 and wth the followng values, reectvely as , , ; , , and , , Smlarly when bandwdth parameters, c values ncreases (c = 0., 0.5, 0.75 and the mean values of the estmated parameter approaches the real parameter values. Smlar results have been obtaned (for N = 200 and 500 as n Table 2 and 3, reectvely. The S.D, MSE and RMSE values reduced as ncreased the bandwdth parameters, c values (c = 0., 0.5, 0.75 and. For nstance, n Table 2, the results shown that the S.D, MSE and RMSE values as , and , reectvely wth α-cuts s 0.2 and bandwdth parameter, c s 0.. When bandwdth parameters, c, ncreased (from 0. to 0.5 to 0.75 and, the S.D, MSE and RMSE values decreases as , , ; , , and , , Table. FSPSSM, N = 00 (ρand 0 ϑ 0 = 0 for γ α-cut c Mean S.D MSE RMSE Scence Publcatons 548

8 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 Table 2. FSPSSM, N = 200 (ρ and 0 ϑ 0 = 0 for γ α-cut c Mean S.D MSE RMSE Table 3. FSPSSM, N = 500 (ρ and 0 ϑ 0 = 0 for γ α-cut c Mean S.D MSE RMSE The same results are obtaned for α-cuts = 0.8 where the S.D, MSE and RMSE values reduced as ncreased of bandwdth parameters, c. Smlarly when value ncreases (N = 00, 200 and 500 the mean values of the estmated parameter become smaller. These values ndcated that under bandwdth parameters, c (0., 0.5, 0.75 and of FSPSSM s effcent. 4. DISCUSSION Snce Heckman (979 ntroduced the sample selecton model, ths model has receved consderable attenton (parametrc, sem-parametrc or nonparametrc and has been used n many applcatons. However, the researchers do not put an effort to nvestgate ths model n terms of uncertanty regardless of ts exstence n the model. Thus, n ths study, we ntroduced the fuzzy concepts hybrd wth the semparametrc sample selecton model. The fuzzy concept s an alternatve framework to solve the problem of uncertantes exstng n ths model, partcularly the relatonshp between the endogenous and exogenous varables. Therefore, t wll dsrupt the ablty and effectveness of the model proceeded to gve the estmated value that can explan the actual stuaton of a phenomenon. These are questons and problems that have yet to be explored and the man pllar of ths study. Therefore, ths model was the frst developed usng fuzzy concept known as the Fuzzy Sem- Parametrc Sample Selecton Model (FSPSSM. Scence Publcatons 549

9 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , CONCLUSION In ths study, we studed the consstency for FSPSSM under normalty assumpton. Subsequent of ths assumpton, the effect of the correlaton between fuzzy varables ( wand % x% and the effect of the correlaton between error terms ( ε% and u% are nvestgated. As a contnuaton from that, consstency n FSPSSM usng the bandwdth parameter as ntroduced by Powell (987 s also studed. A Monte Carlo smulaton s used to examne the consstency for FSPSSM under normalty assumpton. The Monte Carlo smulaton results reveal that consstency depends on bandwdth parameter. When bandwdth parameters, c are ncreased from 0., 0.5, 0.75 and as the numbers of N ncreased (from 00 to 200 and ncreased to 500, the values of mean approaches (closed to the real parameter. Accordng to Schafgans (996, ths ndcated that the FSPSSM s consstent. Through the bandwdth parameter also reveals that the estmated parameter s effcent,.e., the S.D, MSE and RMSE values become smaller as N ncreased. In partcular, the estmated parameter becomes consstent and effcent as the bandwdth parameters approaches to nfnty, c as the number of observatons, n tend to nfnty, n. In ths study, we are focusng only for two area whch are fuzzy concept partcullarly on fuzzy number of semparametrc Sample Selecton Model cons as fuzzy sem-parametrc Sample Selecton Model (FSPSSM and to see the effectveness of the proposed model, the smulaton usng monte carlo s used. Ths paper developed of ths proposed modelng approach, future research work could be emphaszed n several drectons. Apparently, the fuzzy concepts defned n ths study consder the TFN and α-cut method. Therefore, future study could consder other fuzzy numbers whch are more advanced such as S- shaped, bell-shaped. Snce the relatonshp between explanatory varables exsts n the models, the concept of lnear programmng-based method ntroduced by Tanaka et al. (982 and Amr and Tularam (202 could be explored. By dong so, perhaps a deeper understandng of the underlyng structure of the models could be obtaned. Thus, some other mathematcal tools such as optmzaton theory could be explored. The most sgnfcant dea of ths research was to brng the concept of fuzzy nto the selectvty model. In general, ths concept s consdered as a platform to dscover a new dmenson usng these models. Further research could consder development of fuzzy perectve on a new paradgm of selecton model, such as nonparametrc and sem-nonparametrc methods, the propertes and theoretcal parts of selecton model, handlng a weaker assumpton and nvestgatng a curse of dmensonalty usng fuzzy concept. In the development of FSPSSM, fuzzy logc usng rules based method can also be consdered. Ths concept wll lead to produce an output based on lngustcs varables and lngustcs modfer. Hence, the proposed model would be useful n order to compute the uncertantes n the models. Thus, t would be nterestng to fnd whether t s possble to determne the percentages to whch any ecfc uncertan parameters of the models contrbute to the overall uncertanty of the models. In ths study, we have developed Monte Carlo smulatons usng R language programmng. These smulatons could be mproved. Babuska and Verbruggen (996; Chandramohan and Kamalakkannan, 204; Hussen and Nordn, 204; Kareem, et al., 204; Kahtan et al., 204; Srdharan and Chtra, 204 mentoned that modelng of complex systems wll always reman an nteractve approach. Thus, future study could consder the usage of other software packages or programmng languages and ncorporate graphc nterface. In ths way, nformaton such as parameter estmate could be easly utlzed and would be benefcal to the decson makers as well as others nterested partes. These methods could be useful n data mnng, e.g., n credt default analyss, healthcare analyss, securty analyss and agrculture analyss. 6. ACKNOWLEDGMENT The author wshesto thank themnstry of Hgher Educaton, Malaysa for onsorng studes and Unversty Malaysa Terengganu (UMT for the support and encouragement of study at the PhD level 7. REFERENCES Ahn, H. and J. Powell, 993. Sem-parametrc estmaton of censored selecton models wth a nonparametrc selecton mechansm. J. Econometr., 58: DOI: 0.06/ (9390-H Amr, S. and G.A. Tularam, 202. Performance of multple lnear regresson and nonlnear neural networks and fuzzy logc technques n modellng house prces. J. Math. Stat., 8: DOI: /jms Scence Publcatons 550

10 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 Andrews, D.W.K., 99. Asymptotc normalty of seres estmaton for nonparametrc and semparametrc regresson models. Econometrca, 59: DOI: / Babuska, R. and H.B. Verbruggen, 996. An overvew of fuzzy modelng for control. Control Eng. Pract., 4: Bellemare, C., B. Melenberg and A. Soest, Sem-parametrc models for satsfacton wth ncome. Portuguse Economc J., : DOI: 0.007/s z Chamberlan, G., 986. Asymptotc effcency n sem-parametrc models wth censorng. J. Econometr., 32: DOI: 0.06/ ( Chandramohan, K. and P. Kamalakkannan, 204. Traffc controlled-dedcated short range communcaton: A secure communcaton usng traffc controlled dedcated short range communcaton model n vehcular ad hoc networks for safety related servces. J. Comput. Sc., 0: DOI: /jcs Cosslett, S., 990. Sem-Parametrc Estmaton of a Regresson Models wth Sample Selectvty. In: Nonparametrc and Sem-Parametrc Estmaton Methods n Econometrcs and Statstcs, Barnett, W.A., J. Powell and G.E. Tauchen (Eds., Cambrdge Unversty Press. pp: Das, M., W.K. Newey and F. Vella, Nonparametrc estmaton of sample selecton models. Rev. Econom. Stud., 70: DOI: 0./ X Gerfn, M., 996. Parametrc and sem-parametrc estmaton of the bnary reonse model of labour market partcpaton. J. Appled Econometr., : DOI: 0.002/(SICI (99605:3<32::AID-JAE39>3.0.CO;2-K Hardle, W., 990. Appled Nonparametrc Regresson. st Edn., Cambrdge Unversty Press, Econometrc Socety Monograph. Heckman, J.J., 979. Sample selecton bas as a ecfcaton error. Econometrca, 47: DOI: /92352 Hussen, I.S.H. and M.J. Nordn, 204. Palmprnt verfcaton usng nvarant moments based on wavelet transform. J. Comput. Sc., 0: DOI:0.3844/jcs Ichmura, H Sem-Parametrc Least Square (SLS and Weghted SLS Estmaton of Sngle- Index Model. J. Econometr., 58: 7-20 Ichmura, H. and L.F. Lee, 99. Sem-parametrc Least Square Estmaton of Multple Index Models: Sngle Equaton Estmaton. In: Nonparametrc and Sem- Parametrc Estmaton Methods n Econometrcs and Statstcs, Barnett, W.A., J. Powell and G.E. Tauchen (Eds., Cambrdge Unversty Press, Cambrdge. Kabaday, O., Range of medum and hgh energy protons and alpha partcles n NaI scntllator. Am. J. Appled Sc., : DOI: /ajas Kareem, S.H., I.H. Al and M.G. Jalhoom, 204. Synthess and characterzatonof organc functonalzed mesoporous slca and evaluate ther adsorptve behavor for removal of methylene blue from aqueous soluton. Am. J. Envron. Sc., 0: DOI:0.3844/ajes Khan, S. and J.L. Powell, 200. Two-step estmaton of sem-parametrc censored regresson models. J. Econometr., 03: DOI: 0.06/S ( Kahtan, H., N.A. Bakar and R. Nordn, 204. Dependablty attrbutes for ncreased securty n component-based software development. J. Comput. Sc., 0: DOI: /jcs Klen, R. and R. Spady, 993. An effcent semparametrc estmator of the bnary reonse model. Econometrca, 6: DOI: / Lee, M.J. and F. Vella, A sem-parametrc estmator for censored selecton models wth endogenety. J. Econometr., 30: DOI: 0.06/j.jeconom Lola, M.S., A.A. Kaml and M.T.A. Osman, Fuzzy Parametrc of Sample Selecton Model Usng Heckman Two-Step Estmaton Models. Am. J. Appled Sc., 6: DOI: /ajas Markus, F., 998. Sem-parametrc estmaton of selectvty models. Ph.D. Thess, Konstanz Unversty, Konstanz. Martns, M.F.O., 200. Parametrc and sem-parametrc estmaton of sample selecton models: An emprcal applcaton to the female labor force n portugal. J. Appled Econometr., 6: DOI: 0.002/jae.572 Scence Publcatons 55

11 L. MuhamadSafh et al. / Amercan Journal of Appled Scences (9: , 204 Safh, L.M., 203. Fuzzy parametrc sample selecton model: Monte carlo smulaton approach. J. Stat. Comput. Smulat., 83: DOI: 0.080/ Nawata, K., 994. Estmaton of sample selecton bas models by maxmum lkelhood estmator and heckman s two-step estmator. Econometr. Lett., 45: DOI: 0.06/ ( Newey, W.K., J.L. Powell and J.R. Walker, 990. Sem-parametrc estmaton of selecton models: Some emprcal results. Am. Economc Rev., 2: Powell, J. J.H. Stock and T.M. Stoker, 989. Semparametrc estmaton of ndex coeffcents. Econometrca, 57: Powell, J.L., 987. Sem-parametrc estmaton of bvarate latent varable models. Socal Systems Research Insttute. Unversty of Wsconsn- Madson. Rana, M.S., H. Md and A.H.M.R. Imon, A robust modfcaton of the goldfeld-quandt test for the detecton of heteroscedastcty n the presence of outlers. J. Math. Stat., 4: DOI:0.3844/jms Rana, M.S., H. Md and A.H.M.R. Imon, A robust rescaled moment test for normalty n regresson. J. Math. Stat., 5: DOI:0.3844/jms Robnson, P.M., 988. Root-N consstent semparametrc regresson. Econometrca, 56: DOI: /92705 Schafgans, M., 996. Sem-parametrc estmaton of a sample selecton model: estmaton of the ntercept; theory and applcatons. Ph.D. Thess, Yale Unversty, New Haven. Srdharan, K. and M. Chtra, 204. Trust based automatc query formulaton search on expert and knowledge users systems. J. Comput. Sc., 0: DOI:0.3844/jcs Tanaka, H., S. Hayash and K. Asa, 982. Lnear regresson analyss wth fuzzy model. IEEE Trans. Syst. Man Cybernet., 2: Vella, F., 992. Smple tests for sample selecton bas n censored and dscrete choce models. J. Appled Econometr., 7: DOI: 0.002/jae Wtchakul, S., P.S.N. Ayudhaya and P. Charnsethkul, A stochastc knapsack problem wth contnuous random capacty. J. Math. Stat., 4: DOI:0.3844/jms Yen, K.K., S. Ghoshray and G. Rog, 999. A lnear regresson model usng trangular fuzzy number coeffcents. Fuzzy Sets Syst., 06: Zadeh, L.A., 965. Fuzzy sets. Inform. Control, 8: DOI: 0.06/S ( X Scence Publcatons 552