Fuzzy Modified Error Terms of Sample Selection Model on Married Women Participation in Labour Force

Transcription

1 Int. Journal of Economcs and Management 8(1): (2014) ISSN X Fuzzy Modfed Error Terms of Sample Selecton Model on Marred Women Partcpaton n Labour Force A. Yusrna a, S. Shazan b and L. MUHAMAD SAFIIH c a,b Unverst Malaysa Kelantan c Unverst Malaysa Terengganu Abstract Heckman sample selecton model (1979) has been wdely used n theoretcal feld and applcatons feld. However, ths model nvolves uncertantes and ambgutes. To solve these problems, a new approach on error terms of Heckman sample selecton model has been developed that hybrds wth fuzzy concept through trangular fuzzy number. Sample selecton model was made up of two equatons whch were partcpant equaton and wage equaton. Snce there exst uncertanty n both varables and error terms of the model, therefore fuzzy error terms method was appled on the errors to obtan effcent values whch were more relable n a fuzzy envronment. Ths method was repeated twce on the errors of sample selecton model to acqure a strong relatonshp between the varables and errors whch explans uncertanty. The data set were obtaned from Malaysan Populaton and Famly Survey 1994 (MPFS 1994). Mnmum values of error terms ndcates that modfed fuzzy sample selecton model wth fuzzy error terms performs much better when uncertanty and fuzzness exst. Thus, n terms of uncertanty t was found out that the proposed method was more effcent because t explans the data as well as the relatonshp between the varables n the model much better than ts counterpart. Keywords: Uncertanty, sample selecton model, women partcpaton, econometrcs, fuzzy number Correspondng Author: E-mal: yusrna@umk.edu.my Any remanng errors or omssons rest solely wth the author(s) of ths paper.

2 Fuzzy Modfed Error Terms of Sample Selecton Model ntroducton Heckman sample selecton model was ntroduced by Heckman (1979) n order to manage non-random samples. Indvdual n a sample study were chosen wth fxed crtera and non-random from a populaton of study. Sample selecton model was dvded nto two parts: structural part and selecton part. The structural part was where the samples whch were requred and ths defnes the desred crtera. Whle for the second part, t was the reduced form of the non-random samples taken from the structural part. As mentoned n Froelch (2002), t can mprove the attrbute of non-random sample and act as a representatve for populaton relatonshp. Sample selecton model has been wdely used n emprcal studes, for example n labour force and women wage, educaton, technology and healthcare (Bhalotra and Sanhueza, 2002; Seshamn and Gray, 2004; Le, 2005; Madden, 2006). Lews (1974) dscussed on the partcpaton of workng women n labour force and selecton bas n determnng the decson of women to partcpate n labour force n long term. There were two man reasons why selecton bas exst n the model. Accordng to Heckman (1979), observed ndvdual were selected accordng to a set of gven crteras and also due to the acton taken by the analyst. For nstance, n the study of women labour force, the nformaton of famly stablzaton were usually needed hence analyss for repeated observatons can be done. Two-step method (Lola et al., 2009; Newey, 2009) was a common method used to estmate SSM as the applcaton was much easer to estmate the model value (Vella, 1998; Martns, 2001; Le, 2005). Sample selecton bas whch exst n the samples cause nconsstency n the estmaton of the model. However, t can be reduced wth ths method. Inverse Mlls rato (Vella, 1998; Le, 2005) was the error terms n the frst step, probt step whch explans the partcpaton whle the second step, ordnary least square step was only estmated on the partcpaton of ndvdual n the study. Mroz (1984) ntroduced Mroz sequence crtera where only marred women wth complete nformaton were ncluded n the study. Whle the rest wll be removed from the sample study. The samples were reduced because of the reducton of marred women samples whch were unusable. Vella (1998) dscussed on the dfferent types of estmaton n econometrc modellng and selecton bas whch exst n sample selecton model. In hs paper, there were three types of sample selecton model dscussed whch was parametrc, semparametrc and nonparametrc model. Accordng to Vella (1998), Heckman model (1979) manage to solve selecton bas. Puhan (2000) dscussed on selecton bas and the effect of varables of educaton attanment towards the outcome of the study. In ths paper, ndvduals who dd not partcpate n labour force, were removed from the observatons. 105

3 Internatonal Journal of Economcs and Management Martns (2001) dscussed on the partcpaton of marred women n labour force n Portugal by usng crsp parametrc and crsp semparametrc sample selecton model. Le (2005) also dscussed on both sample selecton but on applcatons of man and women partcpaton of labour force n Canada. Marred women partcpaton n labour force were dvded nto two categores, whch were partcpant and non-partcpant. Snce, there were no outcome values for nonpartcpant of women, ths caused selecton bas to occur (Martns, 2001). Solo and Orunsola (2007) also dscussed on women wage n Ngera where they concluded that the number of chldren determne women decson to partcpate n labour force by usng two step estmaton and maxmum lkelhood estmaton. Nevertheless, the research dd not dscussed on uncertanty. Therefore, fuzzy concept n sample selecton model was frst dscussed n Muhamad Safh et.al (2006,2008) and Lola et.al (2009). In the study, parametrc and semparametrc sample selecton was developed through fuzzy concept and appled on marred women partcpaton. Up untl now, only ther research dscussed on the fuzzy applcaton whch exst n the model. Fuzzy set theory was developed by Zadeh (1965) to represent membershp functons n fuzzy system. From ths theory, fuzzy attrbutes were shown n quanttatve values. Mathematcal approach were used to get an approxmaton value when nformaton obtaned were uncertan, ncomplete or naccurate. Ths concept has brough to the ntroducton of fuzzy number whch were wdely used n fuzzy judgment. Fuzzy number was dscussed n Dubos and Prade (1980) where t was used when crcumstances cannot be explan n exact values. From fuzzy numbers, t has expanded to trangular fuzzy numbers. There are many types of fuzzy numbers whch are trangular, trapezum, Gauss and bell-shape, but trangular was the most commonly used (Pedrycz, 1994, Lola et.al, 2009). Trangular number has a left and rght trangular fuzzy number as a support for fuzzy number (Dubos and Prade, 1980). The membershp functon n trangular fuzzy number was much smple as t only have three values and can be reduced to two values for a symmetrcal case (Kao and Chyu, 2002). The purpose of ths paper was to ntroduce a modfed form of sample selecton model based on ts fuzzy error terms that can be used to deal wth hstorcal data whch contaned uncertanty. Sample selecton model wll be modfed wth fuzzy error terms method as ntroduced n Kao and Chyu (2002) as well as fuzzy concept. The frst part of ths paper dscussed on sample selecton model and error terms whle the second part dscussed on fuzzy concept. The new proposed method was developed on the thrd part of ths paper. It was appled on real data of marred women partcpaton n Malaysa. The last part dscussed on ts conclusons. 106

4 Fuzzy Modfed Error Terms of Sample Selecton Model Sample Selecton Model and Error Terms Followng (Martns, 2001; Le, 2005), ths model conssts of two equatons whch were partcpant equaton and outcome equaton and can be defne as follows z = wlc + v = f = l b + z 1 y w u 2 0 z = 0 else y = yz f ^ = 1,, Nh (2.1) where z and y were endogeneous varables, wl and x l were exogeneous varables and β and γ were vector parameters. Bas and nconsstency exst on β estmaton n Equaton (2.1). Whle u and v were the random dsturbances or error terms can be shown n Equaton (2.2). u m m v v p (2.2) 2 u uv c + N 0 v fc : f 0 vuv 1 p The error terms of u and v n both partcpant and outcome equaton were assumed to be normal, ndependent and dentcally dstrbuted..d.,.e. N + ^u, vh. In Equaton (2.2), the exstence of covarate value contrbutes to the relatonshp between other endogenous and exogenous varables n sample selecton model. Snce ths paper focused on the modfed error terms n Equaton (2.2), therefore t was rearranged wth = 1, f, N as shown n Equaton (2.3) and Equaton (2.4). Partcpant equaton error term: 2 y = x lb + u 0 = u + xl b (2.3) u = y -^xlb h 107

5 Outcome equaton error term: Internatonal Journal of Economcs and Management z = wlc + v v = z -^wlch = f = l b + z 1 y x u 2 0 z = 0 else y = yz (2.4) where both error terms were hghly correlated and dentcally dstrbuted as shown n Equaton (2.5) and Equaton (2.6). 2 u + N^0^v, 1hh (2.6) The two estmaton method wdely used n sample selecton model were maxmum lkelhood estmaton (Froelch, 2002) and Heckt two step estmaton (Heckman, 1979). Although maxmum lkelhood estmaton was more effcent than Heckt s estmaton, however the complexty of ths method was qute dffcult to execute for a smple equaton such as sample selecton model (Nawata, 2004; Lola et al., 2009). Therefore, Heckman (1979) ntroduced two step estmaton method to overcome ths problem. Heckt s estmaton was appled n sample selecton model as t was more consstent n terms of estmatng the parameters. Hence, varable analyss can be easly done n complex model. The two steps were probt whch was to estmate the partcpant equaton and least square method to estmate the outcome equaton as shown n Equaton (2.1). Probt step was used to estmate the γ value whch were obtaned from all the observaton of Prob ^z = 1 wh = Ez 6 w@ = U^wlch= Cw ^ lch usng sample = 1, f, N to estmate ct. The frst step ncludes condtonal expectaton as shown n Equaton (2.7): E^y x h = E^y x, z 2 0h= Ex ^ lb x, z 2 0h+ E^u x, z 0h 2 u = xlb+ E^u x, wlc 2-v h (2.7) 2 uv v = xlb+ v v " z^wlch U^wlch, = lb+ nm x 108

6 Fuzzy Modfed Error Terms of Sample Selecton Model From Equaton (2.7), m^ : h = z^wlch U^wlch was the nverse Mlls rato whle z^: h and U^: h were the unvarate probablty dstrbuton and cumulatve dstrbuton functon, wth μ as the covarance between u and v. β, γ and μ can be consstently estmated usng Heckt s two step (Heckman, 1979). The probt model n Equaton (2.7) ncludes probablty dstrbuton functon and was assumed as standard normal cumulatve functon (Horowtz, 2004). However, the second step can be qute complex when the error terms have to be ft wth the frst estmaton (Heckman, 1979; Greene, 1981 and Maddala, 1983). Inverse Mlls rato showed the exstence of bas n sample selecton model. The covarance, v uv between u and v was assume as 1 n probt model to dentfy γ. Snce the two equatons n Equaton (2.1) were seen as two dfferent parts, therefore nverse Mlls rato n probt step was nserted nto least square method as t becomes an entty. Hence, accordng to Muhamad Safh et al. (2011) parametrc estmaton y for n observaton can be consstently estmated and shown as Equaton (2.8): z = wlct + v xlb + v uv 2 z = 1^xlb + u 0h (2.8) where γ parameter n Equaton (2.8) was estmated usng least square method to obtaned consstently estmated parameter. Snce the error terms were hghly correlated wth each other, the regresson varable z on w for selected samples gave nconsstent estmaton. It has been well known that t depends on normal dstrbuton assumpton for an estmaton to be consstent. On the other hand for dentfyng purpose, varable x must contan at least one varable more than varable w (Martns, 2001). Therefore, a new modfed sample selecton model has been developed to handle nconsstency estmaton problem where uncertanty exst. Fuzzy concept The elements n a crsp set were determned by the membershp functon that can be ether 0 or 1 whch s a bnary system. The membershp functon n a crsp set was lmted as t must be execute precsely. However, human percepton were not always precse and ths apples on the surroundng as well. Ths causes human percepton to change often and contans uncertanty (Zadeh, 1965). Element of uncertanty exst to mprove fuzzy mathematc model whch were caused by other factors (Ekel, 2002). Snce mult crtera model such as sample selecton model contans more than one crtera, factor and varable, thus explanaton usng crsp 109

7 Internatonal Journal of Economcs and Management model wll gve naccurate outcome as t contans uncertantes n each of t. Therefore n a fuzzy envronment, a fuzzy approach was more sutable to explaned these values (Zadeh, 1965). Let say x s doman whle x s the element of set P. Therefore, a fuzzy set P u wth the respected membershp functon can be defned as Equaton (3.1). np ^xh: x " where n P ^xh = 1 f all x s n P, n P ^xh = 0 f none of x s n P, 0 1 n ^xh 1 1 f some of x s n P. (3.1) P Fuzzy numbers exst n the error terms of the new hybrd model. The basc propertes of the membershp functon of fuzzy numbers were as follows: (a) A fuzzy number of a fuzzy set s concave and normal (b) Alpha cut for each fuzzy number s n a close nterval of real number confdence level. (c) Each real number n an open nterval (a, d) are real numbers. Fuzzy number whch was trangular fuzzy number was appled on the varables and the error terms of sample selecton model. The bass of trangular fuzzy numbers were as Equaton (3.2): Z f^xh for x! 6a, b@ 1 for x! b, c P x = ] ^ h [ g^xh for x! 6c, d@ ] 0 for x 1 a and x 2 d \ (3.2) wth a # b # c # d. f s a contnuous functon whch ncreases monotoncally from pont b to 1 whle g s a contnuous functon whch decreases from pont c to 1. Whle the membershp functons for trangular fuzzy numbers wth P u were (Kaufmann and Gupta, 1991): Za^q-b m-bh f q! 6b, m@ 1 f q m np q = ] = ^ h [ a^a-q a-mh f q! ] 0 otherwse. \ 110

8 Fuzzy Modfed Error Terms of Sample Selecton Model Two man assumptons for the membershp functon of n ^qh were: 1. np ^qh ncreases monotoncally wth n P ^qh = 0 and lm n q 1 q P ^ h = for " 3 q # x np ^qh decreases monotoncally wth n P ^qh = 1 and lm n q 0 q P ^ h = for " 3 q $ x 2. Defuzzfcaton was used to change the fuzzy output to crsp values. Accordng to Sanefard and Asghary (2011), there were seven method of defuzzfcaton but the most commonly used s centrod method. Ths defuzzfcaton method used crsp values whch contan uncertanty as a mddle value to obtaned fuzzy dstrbuton. Ths s the best method as t ncluded all values n observaton and changes t nto one value only. It s also less-complcated and more effcent n obtanng the defuzzfed values. Uncertanty exsts on the error terms, endogenous varables and exogenous varables of Heckman sample selecton model (Heckman, 1979). There were two types of endogenous varables; the frst was crsp whch have nteger values whle the second was fuzzy whch contan uncertanty. Accordng to Kao and Chyu (2002), a crsp model becomes a fuzzy model f one of the elements n t contans ether vagueness or uncertanty. Therefore, n a fuzzy envronment a crsp exogenous varable transformed from crsp to fuzzy as t nherts the vagueness of the model. In addton, t became fuzzy because of the mxture of crsp and fuzzy elements. To overcome these uncertanty problems n sample selecton model, fuzzy error terms method and fuzzy concept were the basc for development of modfed fuzzy sample selecton model based on ts error terms. P Fuzzy Error Terms of Sample Selecton Model As mentoned before, crsp sample selecton model have two error terms ^u, vh wth each of t was lmted and restrcted due to uncertanty and can be mproved through fuzzy set. The membershp functon of fuzzy set was hybrd nto the model, to obtaned modfed fuzzy error terms. It was also known that both crsp error terms were hghly correlated, normal and ndependently dentcally dstrbuted. Thus, by heredtary the fuzzy error terms also have the same attrbute. From Equaton (2.3) and Equaton (2.4), the error terms whch contans uncertanty were hybrd wth the propertes of fuzzy set n Equaton (3.1), thus a modfed fuzzy error terms were shown as Equaton (4.1) and (4.2). 111

9 Partcpant equaton fuzzy error term: Outcome equaton fuzzy error term: Internatonal Journal of Economcs and Management z = wlc + vu v u = z - ^w l h c y = x lb + u 0 2 = u + xlb u = y -^xlb h (4.1) 2 z = 1 f y = xlb + u 0 (4.2) z 0 = else y = y z where z and y were endogeneous varables, wl and x l were exogeneous varables and β and γ were vector parameters. By Equaton (4.1) and Equaton (4.2), fuzzy error terms dstrbuton of u and vu can be obtaned as n Equaton (4.3) and Equaton (4.4). 2 u + N^0^v, 1hh (4.3) u 2 vu + N^0^v, 1hh (4.4) From Equaton (4.1) and Equaton (4.2), the varables were fuzzy as well as t nherted the attrbute of the model whch contans uncertanty. Therefore, Equaton (4.1) whch was hybrd wth fuzzy error terms method (Kao and Chyu, 2002) were shown n Equaton (4.5). The steps shown were accordng to Kao and Chyu (2002). v u = yu -^xu b h (4.5) wth xu l and y u as fuzzy varables and b as parameter. Let say R = ( b, 0, a) was the estmaton of u wth b s left or lower value and a s the rght or upper value. Thus, the estmaton varable became yut = xu lb + Ru (4.6) 112

10 Fuzzy Modfed Error Terms of Sample Selecton Model The fuzzy observaton value, y ut and fuzzy estmaton yut n trangular fuzzy number form were y = ^y, y, y h and yt = ^yt, yt, yt h respectvely. If D 1 b m a b m a s the estmaton error for u n partcpant equaton of Equaton (4.6), t can be mnmzed as Mn n / D1 = 1 ^ s.t. D1 = # n y y u ^ h-n yu yh dy (4.7) u u t sy jsy It was known that yut = xu l b+ Ru 1 where R1 = ^- b, 0, a h. If b mn and a mn as the least left and rght value, therefore t can be obtaned from ^ytm - ytb h $ bmn, ^yta - ytm h $ amn. The estmaton varable of y ut a were represented as follows yt = ^yt, yt, yt h b m a wth the parameter values respectvely were yut = ^ xl -bh, yut = ^ xl h, yut = ^ xl + ah b b 1 Hence, n smplfed form b yt = ^yt, yt, yt h b m a m b 1 m = ^b xl - b, b xl, b xl + ah (4.8) 1 b 1 m 1 a = ^b + b xl - b, b + b xl, b + b xl + ah a b b 0 1 m 0 1 a As for the th observaton, the dfferences of were calculated as Equaton (4.9): a D u 1 = 05. ^ ya- ybh a+ b+ b1 ^ xa l -xb l h@ -205 ^. h6y - b + b ^xl -bh@ a a 0 1 b 1 (4.9) where a 1 = heght^yu k yut h = ^y -yt h ^y - y + yt -yt h a b a m m b 113

11 Internatonal Journal of Economcs and Management as a 1 value was the membershp functon of the y ut and y ut ntersecton. In ths study, the alpha cut values were determned based on the experence and observaton done by the fuzzy expert. After the frst D Du 1 = u was obtaned, t was nserted back nto Equaton (4.5) to obtaned Equaton (4.10): Du = u = yu - ^xu l h (4.10) 1 b Centrod method for defuzzfcaton was used the fnd the crsp values of the fuzzy error terms and varables. If u m, y m and x m l were the defuzzfy values of u, y u and xu l, therefore the crsp values were: 3 3 m = - u u 3-3 u un ^udu h n ^uhdu 3 3 # #, y = # yn ^ydy h n ^yhdy 3 # and # # 3 m l xl - -3 xl m - y -3 y 3 x = n ^xhdx n ^xdx h 3 Wth all the observatons of u m, y m and xl m became u = 1 3^u + u + u h, y = 1 3^y + y + y h m b m a and xl = 1 3^xl + xl + xl h m b m a m b m a Snce symmetrcal fuzzy number was used on ths study, therefore the crsp values of the error terms and varables were the mddle values whch were u m, y m and x m l respectvely. Through ths defuzzfcaton method, Equaton (4.10) were rearranged so that crsp values of partcpaton equaton were obtaned as follows: D1 = u = y - ^xl bh u = y -^xl bh (4.11) y = u + xlb The steps from Equaton (4.5) untl Equaton (4.11) were repeated once on the outcome equaton n Equaton (4.2) to obtaned the second estmaton error value, D2 = v. Thus, fuzzy error term method on the second equaton was shown n Equaton (4.12). 114

12 Fuzzy Modfed Error Terms of Sample Selecton Model D2 = v = z - ^wl ch v = z -^wl ch (4.12) z = wlc + v A new modfed fuzzy sample selecton model wth fuzzy error terms has been developed and were shown n Equaton (4.13) f else zu = wul + vu 1 c z u = 1 f yu = xul b + u 2 0 (4.13) z = 0 else yu = yu zu (=1,..., N) where z u and were fuzzy endogenous varable, wu l and xu l were fuzzy exogenous varable, β and γ were the parameters. Endogenous varables were observable whle exogenous varable were unobservable. Ths new modfed model were appled on real data set of marred women partcpaton n Malaysa. Data Set of Sample Selecton Model Modfcaton The data set whch was used on ths study were obtaned from the Malaysan Populaton and Famly Survey 1994 (MPFS-94), whch provdes nformaton on wages, educatonal attanment, household composton and other socoeconomc characterstcs of marred women n Malaysa. The survey data were from 1994 and have been provded by Natonal Populaton and Famly Development Board of Malaysa under Mnstry of Women, Famly and Communty Development Malaysa. The orgnal sample data contaned 4444 samples but through sequental crtera (Mroz, 1984) t was reduced to 2792 sample data as the rest of the data was ether ncomplete or have no recorded famly ncome n 1994 (Lola et al. 2009). Therefore, 1100 (39.4%) were marred women wth partcpants n labour force whle 1692 (60.6%) were non partcpants. The analyss were lmted to women marred and aged below 60, not n school or retred, husband present n 1994 and husband reported postve earnng for 1994 based on selecton rules (Martns, 2001) whch was appled to create the sample crtera n choosng for partcpant and non partcpant of marred women n MPFS-94 data set. 115

13 Internatonal Journal of Economcs and Management W Z = b + b A+ b A + b EDU + b CHD+ b H + u G = c + c EDU + c P + c P + c P + c P + v p EXP 3 EXP2 4 EXPCHD 5 EXPCHD2 Z 1 = f Outcome ^Yh = (5.1) Z 0 = else y G p = c + c EDU+ c P + c P + c P EXP 3 EXP2 4 EXPCHD + c P + v 5 EXPCHD2 = G Z = 1, f, N p From Equaton (5.1), the endogenous varables n modfed fuzzy sample selecton model werez and G p, whle the exogeneous varables were AGE (A, age n a year dvded by 10), AGE2 (A 2, age square dvded by 100), EDU (years of educaton), CHD (number of chldren under 18 lvng n the famly) and HW (H W, log of monthly husband s wage). β and γ were unknown parameters, whle u and v were the error terms of sample selecton model for sample data for women. It was also hghly correlated, normal, ndependent and dentcally dstrbuted wth each other as shown n Equaton (2.2). For the determnaton of wages, standard human captal approach was followed except for potental experence whch was calculated age-edu-6 nstead of actual work experence wth Buchnsky (1998) soluton was adopted. Snce there were no data for real workng experence, therefore potental wage was obtaned from potental experence, G p (Muhamad Safh et al. 2008). When partcpaton = 1 Z ^ 2 0 h, therefore outcome equaton becomes 2 Gn = 1EXP + 2EXP wth EXP as real workng experence for non-partcpant. Sample selecton model n Equaton (5.1) were rearranged n order to calculate the error terms as stated n Equaton (5.2) and (5.3). Followng the justfcaton by Kao and Chyu (2002), sample selecton model were also classfed as fuzzy. Therefore, Partcpaton equaton error terms: Z u = b0+ b A u 1 + b2a u 2+ b3edu + b4chd+ b5h u W + u u = Zu - ^b + b Au + b Au + b EDU+ b CHD + b Hu h (5.2) W 116

14 Outcome equaton error terms: Fuzzy Modfed Error Terms of Sample Selecton Model Gu p = c1edu + c Pu 2 EXP + c3pu EXP2+ c4pu EXPCHD+ c 5 Pu EXPCHD2+ vu vu = Gu - ^c + c EDU+ c Pu + c Pu + c Pu + c Pu h (5.3) p EXP 3 EXP2 4 EXPCHD 5 EXPCHD2 Thus, the modfed sample selecton model wth fuzzy error terms for sample data of women were shown n Equaton (5.4). Zu = b + b Uu+ b Uu + b EDU + b CHD+ b Gu + D S 1 Gu = c0+ c1edu + c2pu + c3pu 2+ c4pu + c Pu + D p EXP EXP EXPCHD 5 EXPCHD2 2 Z = 1 f Outcome ^Yh= G u p = c0+ c1edu+ c2pu + c3pu 2+ c4pu + c Pu + D Z 0 = else y 5 EXPCHD2 2 EXP EXP EXPCHD = G Z = 1, f, N p (5.4) In Equaton (5.1), there were two types of varables whch contan uncertanty and nteger lkewse. It s known that the data contan uncertanty, thus trangular fuzzy number s hybrd nto all of the exogenous and endogenous varables except for EDU and CHD, whch were crsp. The fuzzy endogenous varables n ths study wer age, wage and potental experence. Snce there were some parts of sample selecton model contans uncertanty, therefore nteger values n exogenous varables namely EDU and CHD were categorze as fuzzy varables (Kao and Chyu, 2002). There were two fuzzy endogenous varables, frst was bnary varables n partcpaton equaton and log hourly wages (HW) n outcome equaton. The bnary varables were made up of 2 ndcators where 1 represented partcpant and 0 as non-partcpant. Accordng to Lola et al. (2009), non-partcpant women were self-employed ether n famly busness, farmng or a full-tme housewfe. Whle as for fuzzy exogenous varables, they were n both partcpaton and outcome equaton. AGE was n partcpant equaton, PEXP was n outcome equaton whle EDU contaned n both equaton. The man functon of AGE and EDU were to determne human captal and were expected to have negatve probablty to be hred n labour force (Lola et al. 2009). 117

15 Internatonal Journal of Economcs and Management The summary of varables used n ths study and further dscusson on the data varables can be referred n Muhamad Safh et al. (2008). Sample sze for both crsp and modfed sample selecton model was N = The crsp sample selecton was calculated wthout any changes made to the data. Whle the fuzzy sample selecton model was modfed towards error terms, exogenous and endogenous varables at dfferent alpha cut values. Comparson was made between the values of partcpaton and outcome for coeffcent estmaton, sgnfcaton and error terms aganst partcpaton equaton and outcome equaton for both models. The emprcal result for both models wth the respectve standard error (n bracket) was shown n Table 5.1 and Table 5.2. Table 5.1 Parametrc estmaton of sample selecton model and modfed fuzzy sample selecton for partcpant equaton Partcpaton Equaton Varables/ Alpha Cut () Sample Selecton Model Constant (0.7122) AGE (0.4218) AGE (0.0535) EDU (0.0092) CHD ( ) HW ( ) at 5%level of sgnfcance () Modfcaton of Fuzzy Sample Selecton Model α = 0.2 α = 0.4 α = 0.6 α = (0.7351) (0.4357) (0.0551) (0.0094) ( ) (0.2819) (0.7300) (0.4325) (0.0547) ( ) ( ) (0.2795) (0.7244) (0.4292) (0.0543) ( ) ( ) (0.2768) (0.7185) (0.4256) (0.0539) ( ) ( ) (0.2738) Table 5.1 showed the emprcal result obtaned from the frst step of Heckt s two step estmaton, whch was probt for partcpaton equaton (Vella (1998), Le (2005) and Martns (2001)). Column () was the crsp sample selecton model result whle column () was the modfed sample selecton model result at alpha cut values of 0.2, 0.4, 0.6 and 0.8. In ths study, the probablty of marred women partcpaton n labour force was assumed by probt model. It was known that the 118

16 Fuzzy Modfed Error Terms of Sample Selecton Model age of women partcpaton was a quadratc functon, therefore AGE and AGE2 varables were hghly sgnfcant wth whch was stated n Al-Quds (1996) artcle. For example, n AGE varable every ncrement n the probablty of marred women jonng the labour force, there would be a steady ncrement n a small magntude. In the crsp sample selecton model, when the probablty n AGE varable changes from 0 to 1 t causes a 54.5% (Pr Y = 1) or 0.545) change n probablty value that wll lead to marred women partcpaton n labour force. As the alpha cut value ncreases from 0.2 to 0.8, the probablty values were 53.5 % (Pr Y = 1) or 0.535), 53.9% (Pr Y = 1) or 0.539) 54.2 % (Pr Y = 1) or 0.542) and 54.4 % (Pr Y = 1) or 0.544) when uncertanty exsts. Ths shows that as women s age ncreases to a certan years, they were more lkely to partcpate n labour force. The probablty estmaton of EDU and CHD were both postve and sgnfcant n the crsp model. Here, the smlar crcumstance were also shown n fuzzy model when the uncertanty exst. It s known that educaton have a postve mpact on marred women s decson to partcpate n labour force. From Table 5.1, 1% change of marred women wth lower educaton partcpate n labour force only n a small rato of 5.23 % ((Pr Y = 1) or ) as n marred women partcpant n Ngera. The partcpaton probablty of marred women n labour force was more frmed when there exst uncertanty for educaton whch was 5.23 % ((Pr Y = 1) or ) and ths value was mantan for alpha values 0.2 to 0.8. Ths outcome was smlar as the justfcaton from Solo and Orunsola (2007) fndngs. Marred women partcpaton n labour force whom have a stable number of chldren (number of chldren 3) has the probablty value of 1.1 % (Pr Y = 1) or 0.011). In the modfed model, there was only a small change n the probablty for number of chldren as alpha value ncreases. Ths shows that the number of chldren a marred women has, does not qute effect ther decson to partcpate n the labour force. A negatve or nsuffcent of husband s wage n supplyng for the famly also effects women decson to partcpate n labour force as mentoned n women s study n Ngera and Portugal. Although the probablty estmaton n crsp model for HW varable was negatve, t was sgnfcant as the result study n Lola et al. (2009) and ths also apples for the modfed model. A 1% probablty change n husband wage, the probablty of marred women partcpaton decreases to 22 % ( Pr Y = 1) they were 21.3 % (Pr Y 1) or 0.22). Whle for the modfed model, = or 0.213), 21.7 % (Pr Y = 1) or 0.217), 21.9 % ( Pr Y = 1) or 0.219) and 22.2 % (Pr Y = 1) or 0.222) whch shows small changes as alpha ncreases. The probablty values obtaned, showed the uncertanty nsde the model whch were not shown by the crsp model. 119

17 Internatonal Journal of Economcs and Management It was found out that, all the fuzzy varables were consstent although the estmaton value ncreases from 0.2 to 0.8. However, t was stll approachng to the estmaton probablty of the crsp model whch shows mnmum values of the model. Thus, probt step of partcpaton equaton shows that the modfed sample selecton model were more effcent n dealng wth uncertantes. Table 5.2 Parametrc estmaton of sample selecton model and modfed sample selecton model for wage equaton. Wage Equaton Varables/Alpha Cut () Sample Selecton Model Constant (0.1039) EDU (0.0019) PEXP (0.0574) PEXP (1.3095) PEXPCHD (0.0117) PEXPCHD (0.0037) at 5%level of sgnfcance () Modfcaton of Fuzzy Sample Selecton Model α = 0.2 α = 0.4 α = 0.6 α = (0.0210) (0.0004) (0.0117) (0.2702) (0.0025) (0.0008) ( ) (0.0007) (0.0231) (0.5296) (0.0048) (0.0015) (0.0623) (0.0011) (0.0344) (0.7885) (0.0070) (0.0022) (0.0829) (0.0015) (0.0458) (1.0461) (0.0093) (0.0029) Table 5.2 shows the emprcal result of least square method gven by the second estmaton of outcome equaton. In column () was the crsp sample selecton model result whle column () was the modfed sample selecton model result at alpha cut values of 0.2, 0.4, 0.6 and 0.8. The coeffcent values obtaned were sgnfcant wth postve values of PEXP and PEXPCHD2 varables, whle the others were negatve. Marred women wth workng experences have changes n ther ncome/ wage of 5.84% ( x 100%) and can reached up to 34.39% ( x 100%) wth each changes of 1% n PEXP when there was uncertanty n the model. The least square results show that marred women ncome ncreases dependng on ther workng experences as stated n Phmster (2004). Ths fndng was clarfed wth the ncrement of women s ncome as alpha approaches 1 n the modfed model. The results for potental experence (PEXP and PEXP2) were sgnfcant as the fndngs obtaned from Martns (2001). Although negatve coeffcents were obtaned for PEXPCHD and PEXPCHD2 varables, t was n lne wth the economc 120

18 Fuzzy Modfed Error Terms of Sample Selecton Model theory as stated n Coelho et al. (2005). It was also found out that educaton does have a negatve relatonshp wth women s ncome whch was the same as the study n Ngera. For each 1% ncrement of educaton of marred women, t reduces ther ncome to 0.20% ( x 100%). In fuzzy envronment, women s ncome declne between 0.97% ( x 100%) up tll 1.2% ( x 100%) wth every 1% of change n educaton attanment whch supports the value obtaned n the crsp model. Ths stuaton occurs probably due to the lack of sutable occupaton n certan parts of Malaysa, such as rural areas. It apples to those who have hgher educaton whch was also dscussed n the study of marred women n Ngera. The modfcaton sample selecton model result for wage equaton shows that the error terms n the fuzzy coeffcent were much smaller than the error terms n the crsp coeffcent. In the modfed model, there were only small changes found on the coeffcent estmaton as the alpha cut values ncreases from 0.2 to 0.8. Although the fuzzy coeffcent values were spreadng from crsp coeffcent, nevertheless the values of the error terms was smaller n the modfed model compared to the crsp model. Ths shows a strong relatonshp between the fuzzy varables n sample selecton model when there was uncertanty. Null hypothess test were also done wth zero correlaton (ρ = 0) at 5% sgnfcant level n both crsp and modfed sample selecton model. In the partcpaton equaton, famly sze and husband s wage were faled to be rejected at 5% sgnfcant level whch shows that both factor rely to each other n women s decson to partcpate n labour force. These fndngs were smlar to the study of marred women n Canada, Portugal and Ngera. Whle as for the wage equaton, all PEXP, PEXP2, PEXPCHD and PEXPCHD2 varables were also faled to be rejected at 5% sgnfcant level wth smaller value of ρ (less than 0.05) whch was smlar to Phmster (2004) study. Ths shows that the varables wth mnmze error terms n the wage equaton were sgnfcant towards women s ncome as stated n prevous study. Goodness ft test, R 2 were also done on both crsp and modfed model, whch shows the exstence of bas n the model. The smallest R 2 value was obtaned n the crsp model. Whle as for the modfed model, R 2 value ncreases as alpha cut value ncreases from 0.2 to 0.8. Ths shows that the sample data of marred women used were compatble wth the modfed model whch was smlar to the fndngs n marred women study n Ngera. The sample selecton model only shows the crsp part. However, the fuzzy varables n the modfed sample selecton model whch contans uncertanty performs much better and were much relable than the crsp model. Furthermore, the results for modfed model proves to be more effcent, sgnfcant and wholly n explanng fuzzy and vagueness. Thus, the modfed model was the best alternatve n explanng uncertantes that exst n a model. 121

19 Internatonal Journal of Economcs and Management Concluson From the fndngs, t can be concluded that uncertanty can be explaned more effcently by usng fuzzy approach. A new modfed model was developed to explaned sample selecton model whch has suffered prevously from the defcency through the use of crsp method. In ths study, uncertantes were explaned by mnmzng the error terms as well as varables of the sample selecton model. Snce the sample data of marred women used were hstorcal data, crsp method were not sutable to explaned these data as t the error terms, endogenous and exogenous varables contans uncertanty. Fuzzy varables whch were used n sample selecton model becomes more effcent as uncertanty and vagueness can be explaned through these fuzzy methods. The whole observaton of marred women partcpaton n labour force can be seen more throughly compared to the crsp approach. Thus, through modfed fuzzy sample selecton model, uncertanty can be reduced and sample selecton model performs more effcently where there exst vagueness. Not only modfcaton of sample selecton model successfully explaned uncertanty, but t also manage to gve a sgnfcant contrbuton towards selecton models and econometrc models. The mnmze error terms causes the expert to understand the relatonshp between the varables n the sample data much better. Fnally, snce the results obtaned were more accurate, therefore t can help the economy model legslaton to mprove the new polcy especally for marred women wage n Malaysa n the future. REFERENCES Al-Quds, S.S. (1996) Labor partcpaton of Arab women: estmates of the fertlty to labor supply lnk, Economc Research Forum. Bhalotra, S. & Sanhueza, C. (2002) Parametrc and sem-parametrc estmatons of the return to schoolng n South Afrca, Oxford Unversty. Buchnsky, M. (1998) The dynamcs of changes n the female wage dstrbuton n the USA: A quantle regresson approach. Journal of Appled Econometrcs, 13: Coelho, D., Vega, H. & Veszteg, R. (2005) Parametrc and semparametrc estmaton of sample selecton models: an emprcal applcaton to the female labour force n Portugal, UFAE and IAE workng papers , Untat de Fonaments de l Anals Economca (UAB) and Insttut d Analss Economca (CSIC). Dubos D. & Prade, H. (1980) Fuzzy sets and systems: Theory and applcatons, Academc Press, New York. Ekel, P. Y. A. (2002) Fuzzy sets and models of decson makng, Computers and Mathematcs wth Applcatons 44: Froelch, M. (2002) Semparametrc estmaton of selectvty models, Nova Scence Publshers, New York. 122

20 Fuzzy Modfed Error Terms of Sample Selecton Model Greene, W.H. (1981) Sample selecton bas as a specfcaton error: comment. Econometrca 49(3): Heckman, J. (1979) Sample selecton bas as specfcaton error, Econometrca, 47: Horowtz, J.L. (2004) Semparametrc models. In Handbook of computatonal statstcs, ed. J.E. Gentle, W. Hardle, & Y. Mor. Sprnger: New York. Kao, C. & Chyu C.L. (2002) A fuzzy lnear regresson model wth better explanatory power. Fuzzy Sets and Systems 126: Kaufmann, A. & Gupta, M. (1991) Introducton to fuzzy arthmetc-theory and applcatons, Van Nostrand Renhold, New York. Le, J. J. (2005) Parametrcs and semparametrc estmatons of the return to schoolng of wage workers n Canada, Smon Fraser Unversty. Lews, H. G. (1974) Comments on selectvty bases n wage comparsons. Journal of Poltcal Economy 82: Lola, M. S., Kaml, A. A. & Abu Osman, M. T. (2009) Fuzzy parametrc of sample selecton model usng Heckman two step estmaton models, Amercan Journal of Appled Scences 6(10): Madden, D. (2006) Sample selecton versus two-part models revsted: the case of female smokng and drnkng, Health, Economcs and Data Group, Workng paper 06/12, Unversty College Dubln. Maddala, G. S. (1983) Lmted-dependent and qualtatve varables n econometrcs, Vol 1 USA: Cambrdge Unversty Press. Martns, M.F.O. (2001) Parametrc and semparametrc estmaton of sample selecton models: An emprcal applcaton to the female labour force n Portugal, Journal of Appled Econometrcs 16: Mazumdar, D. (1991) The urban labor market and ncome dstrbuton: A study of Malaysan, Oxford Unversty Press, New York, Mroz, T.A. (1984) The senstvty of an emprcal model of marred women s hours of work to economc and statstcal assumptons. Ph.D. dssertaton, Stanford Unversty London, UK. Muhamad Safh, L., Basah Kaml, A. A. & Abu Osman, M. T. (2006) Fuzzy approach to sample selecton model, WSEAS Transactons on Mathematcs, Issue 6(5): Muhamad Safh, L., Basah Kaml, A. A. & Abu Osman M. T. (2008) Fuzzy sem-parametrc sample selecton model case study for partcpaton of marred women, WSEAS Transactons on Mathematcs Issue 3(7): Muhamad Safh, L. (2011) Fuzzy parametrc and fuzzy semparametrc sample selecton model, Thess submtted for the Degree of Phlosophy (Scence), Unversty Sans Malaysa, Malaysa. Nawata, K. (2004) Estmaton of the female labor supply models by Heckman s two-step estmator and the maxmum lkelhood estmator, Mathematcs and Computers n Smulaton 64:

21 Internatonal Journal of Economcs and Management Newey, W. K. (2009) Two-step seres estmaton of sample selecton models. The Econometrcs Journal, 12(s1), S217-S22. Pedrycz, W. (1994) Why trangular membershp functons? Fuzzy Sets and Systems 64: Phmster, E. (2004) Urban effects on partcpaton and wages: are there gender dfferences? Centre for European Labour Market Research, Department of Economcs, Unversty of Aberdeen, Dscusson paper Puhan, P. A. (2000) The Heckman correcton for sample selecton and ts crtque, Journal of Economc Surveys 14: Sanefard, R. & Asghary, A. (2001) A method for defuzzfcaton based on probablty densty functon (II), Appled Mathematcal Scence Vol 5(28): Seshamn, M. & Gray, A. (2004) Ageng and health care expendture: The red herrng argument revsted, Health Economcs Vol 13: Solo, O. & Orunsola, E. O. (2007) Sample selecton analyss of wage equaton for women n Ondo state Ngera, Journal of Socal Scences 3(3): Vella, F. (1998) Estmatng models wth sample selecton bas: A survey, The Journal of Human Resources; 33: Zadeh, L.A. (1965) Fuzzy sets, Informaton and Control 8: