TheStableBoundedTheoryanAlternativetoProjectingPopulationsTheCaseofMexico

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1 Global Journal of Managemen and Business Research: G Inerdisciplinary Volume 8 Issue 3 Version.0 Year 08 Type: Double Blind Peer Reviewed Inernaional Research Journal Publisher: Global Journals Online ISSN: & Prin ISSN: The Sable Bounded Theory an Alernaive o Projecing Populaions: The Case of Mexico By Javier González-Rosas & Iliana Zárae-Guiérrez Naional Populaion Council Absrac- Nowadays he populaion daa of counries as Japan, India, China, Unied Saes and Mexico, a glance seem o evolving over ime according o a logisic paern. In his conex arises he following research quesion: will here be any form o prove he hypohesis of he logisic paern? Bu his quesion implies hree quesions more, is here exis a minimum and a maximum for populaion growh? Will be able o be he values of he maximum and minimum deermined numerically? And how can his informaion be used o projecing he populaion? In order o answer above quesions we use he Sable Bounded Theory. The daa we used in his paper were elaboraed by Naional Insiue of Saisic and Geography from Mexico and hey cover las 5 years. Key resuls of he paper indicae ha; firs, in Mexico he assumpion abou he logisic paern is rue, second, minimum value for populaion growh of Mexican populaion is 7. million, while maximum is 53.6; and hird, using he minimum and maximum values esimaed and he Logisic paern we forecased México s populaion, so ha, in 00 will be 5.8 million, in 030 will be 34.5 million, for 040 will be 4., and in 050 will i arrive o Keywords: forecas, populaion, sabiliy, logisic paern, gaussian paern. GJMBR-G Classificaion: JEL Code: C53 TheSableBoundedTheoryanAlernaiveoProjecingPopulaionsTheCaseofMexico Sricly as per he compliance and regulaions of: 08. Javier González-Rosas & Iliana Zárae-Guiérrez. This is a research/review paper, disribued under he erms of he Creaive Commons Aribuion-Noncommercial 3.0 Unpored License hp://creaivecommons.org/licenses/by-nc/3.0/), permiing all non-commercial use, disribuion, and reproducion in any medium, provided he original wor is properly cied.

2 The Sable Bounded Theory an Alernaive o Projecing Populaions: The Case of Mexico Javier González-Rosas α & Iliana Zárae-Guiérrez σ Absrac- Nowadays he populaion daa of counries as Japan, India, China, Unied Saes and Mexico, a glance seem o evolving over ime according o a logisic paern. In his conex arises he following research quesion: will here be any form o prove he hypohesis of he logisic paern? Bu his quesion implies hree quesions more, is here exis a minimum and a maximum for populaion growh? Will be able o be he values of he maximum and minimum deermined numerically? And how can his informaion be used o projecing he populaion? In order o answer above quesions we use he Sable Bounded Theory. The daa we used in his paper were elaboraed by Naional Insiue of Saisic and Geography from Mexico and hey cover las 5 years. Key resuls of he paper indicae ha; firs, in Mexico he assumpion abou he logisic paern is rue, second, minimum value for populaion growh of Mexican populaion is 7. million, while maximum is 53.6; and hird, using he minimum and maximum values esimaed and he Logisic paern we forecased México s populaion, so ha, in 00 will be 5.8 million, in 030 will be 34.5 million, for 040 will be 4., and in 050 will i arrive o Keywords: forecas, populaion, sabiliy, logisic paern, gaussian paern. I. Inroducion T he forecas of Mexico's populaion have been radiionally done using he demographic componens mehod, which is based on he esimaion of birhs, deahs, migrans and immigrans, which are elemens ha deermine he change in human populaions. This mehod esimaes birhs and deahs, projecing radiionally feriliy and moraliy using logisic funcions (Parida, 006). However, emigrans and immigrans have no been projeced using mahemaical models; heir forecas has been resriced only o esablishing assumpions abou heir behavior in fuure. The demographic componens mehod also provides us demographic dynamics of he counry hrough predicing he fuure behavior of is componens such as feriliy, moraliy and inernaional migraion. Due o his, he mehod can inroduces for each componen (feriliy, moraliy and migraion) several sources of error such as: ) Errors in daa, since if he daa are no reliable or accurae, hey will produce biased forecass; ) Logisic funcions used o projecing moraliy and feriliy are ofen no adequae, and 3) The minimum and maximum ha are se in order Auhor α σ: Naional Populaion Council from Mexico. s: xavier.gonzalez@conapo.gob.mx, izarae@conapo.gob.mx o projecing feriliy and moraliy are no saisical esimaions, bu are fixed in an ambiguous way. So ha, if all above error ypes are presen in he hree componens, nine error sources would be inroduced. Bu if is projeced he oal populaion only is inroduced hree error sources and so, he projecions may be more accurae. So, purpose of his paper is firsly, demonsrae ha he populaion growh in Mexico follows mahemaical laws very accurae ha allows esimae he maximum and minimum of he growh and besides deermine he evoluion paern of Mexico s populaion hrough ime, and secondly, use he resuls o obain forecass of he Mexican populaion unil year 050. II. Lieraure Review The Unied Naions (UN) publishes projecions of populaions around he world. Tradiionally, The UN produced hem wih sandard demographic mehods based on assumpions abou fuure feriliy raes, survival probabiliies, and ne migraion. Such projecions, however, were no accompanied by formal saemens of uncerainy expressed in probabilisic erms. In July 04 he UN for he firs ime issued official probabilisic populaion projecions for all counries o 00 (Alema, 05). There exis several mehods o projec he populaion. Some of hem projec he oal populaion using an iniial populaion and fuure raes of populaion growh. Oher which is called componens mehod, projecs he populaion by age and sex using an iniial age and sex srucure of he populaion and projecions of feriliy and moraliy (The Cohor Componen Mehod for Maing Populaion Projecions, 07). Some inernaional organizaions prepare populaion projecions for he world, regions and counries. One of hem organizaions is he UN and he U.S. Census Bureau. Oher organizaions as World Ban and he Inernaional Insiue for Applied Sysem Analysis (IIASA) also prepare populaions projecions for world, major regions, and for individual counries. Each of hese organizaions uses slighly differen mehodologies, maes assumpions also differen abou he fuure demographic rends, and begins wih slighly differen esimaes of curren populaion size. Neverheless, for he nex 50 years heir resuls fall wihin a relaively small band (Populaion Reference Bureau, 07). 08 Global Journals

3 According o he World Populaion Prospecs: he 05 revision, nowadays world s populaion is 7.3 billion and is expeced o reach 8.5 billion in 030, 9.7 in 050, and. in 00. China and India coninue being he wo counries wih more populaion in he enire world, represening 9 and 8% of he world s populaion respecively. However, projecions indicae by 0 he India s populaion will be greaer han he China s populaion. Today, one of he en counries ha have more populaion worldwide is in Africa (Nigeria), five are in Asia (Bangladesh, China, India, Indonesia and Paisan), wo are in Lain America (Brazil and Mexico), one is in Norhern America (USA) and one is in Europe (Russian Federaion) (Unied Naions, 07). Currenly, he world populaion coninues o grow alhough more slowly han in he recen pas. Ten years ago, world populaion was growing by.4 per cen per year. Today is growing by.8 per cen per year or approximaely an addiional 83 million people I is very imporan poin ou ha he mos daa of he able were calculaed by a populaion census. b) Evoluion of Mexico s populaion a las 5 years A las 5 years Mexico s populaion has grown under effec of condiions socials, poliics and economic very differen. Through his period we can idenify hree scenarios ha undoubedly have conribued o esablish he demographic dynamic and he populaion volume ha has he counry nowadays. The firs scenario is locaed in he nineeenh cenury in which he growh of he populaion was very slow. The second scenario refers o a par of he wenieh cenury in which growh coninued slowly, bu from he year 930 began an exponenial growh. The hird scenario is locaed a end of he 0h cenury and wha goes of cenury XXI, in which he exponenial growh of he populaion has ended o giving pass o slower growh han he exponenial (Figure ). Table : The populaion in Mexico, Year Populaion Year Populaion Source: INEGI, 07a; 05 INEGI 07b annually. The mos demographers worldwide expec his growh will coninue during he res of his cenury (World Populaion Hisory, 07). Also is very imporan consider ha a projecion is no a predicion abou wha i will happen, i is indicaing wha will happen if he assumpions which underpin he projecion acually occur. These assumpions are ofen based on paerns and daa rends which we have previously observed (Ausralian Bureau of Saisics, 07). III. Mehodology a) Daa used Daa we used in his paper were Mexico s populaion of he las 5 years and hey covered period. Source of hese daa is he Naional Insiue of Saisic and Geography (INEGI by is acronym in Spanish) (Table ). 08 Global Journals

4 Million people Source: Table Figure : The populaion in Mexico, As we can see in he figure, in 790, Mexico had only 4.64 million of inhabians and had o passing lile more han 90 years for he populaion reached he double. For 900, he populaion was 3.6 million of persons, while in 960 reached 34.9 million, his is, 7.7 million more han he double of 900, his means ha, a process ha oo more han 90 years in cenury XIX, in cenury XX i oo a lile more han 50 years. Bu he equal o a quaniy deermined by he mahemaical funcion plus a cerain random deviaion which i will happen according a probabilisic law. Medhi (98) called o he mahemaical funcion, he deerminisic componen and he random deviaion, he sochasic componen. Such ha, under hese posulaes he behavior equaions of he observaions and he mean of he populaion in each ime would be: rapid growh of he populaion in cenury XX coninued P = f ( ) + ε () o increasing, so ha, in 980 Mexico reached 66.8 million, only 3 million less han double of 960, wha P = f () indicaed ha in a few more of 30 years he populaion Where, would duplicae again. However, since 980 o 05 PP have passed 35 years and he populaion has no, Denoes populaion in ime, duplicaed, wha seems o indicaing he rapid growh of ff(), Is an unnown mahemaical funcion, he populaion has been sopped. We can also see in figure ha Mexico s populaion has been growing since 5 years ago ℇ Are random variables ha we suppose independens, wih disribuion law Normal, mean μμ ℇ = 0, And consan variance σσ ℇ, and coninuously, so ha arise follow quesion: will coninue μμ pp, Denoes populaion's mean in ime. growing and growing in fuure? We hin no, and alie a lo of demographers in our counry and in worldwide we In order o proving ha value maximum exiss expec he populaion will sabilize or reach a maximum he Sable Bounded Theory uses he populaion s value and will sar o decreasing. In boh cases implies change amoun respec ime. Due o populaion s ha i mus exis a maximum value o populaion's grow change amoun beween a ime and oher is measured and he answers regarding is exisence and he wih he slope of he sraigh line ha joins wo poins of calculaion of his value seem o being in he Sable he bi-dimensional space defined by ime and he Bounded Theory (Gonzalez-Rosas, 0). populaion, we calculaed he slopes and middle values of wo consecuive populaion values of following way: c) The maximum of he Mexican populaion P P P The Sable Bounded Theory ress in wo i + i = i (3) fundamenal posulaes, firs, ha in each year he i+ i populaion is a random phenomenon, so, according o P P i i he probabiliy heory in each year mus have a mean MV P + = P i + (4) i and a variance. Second, he mean of he populaion is equal o a mahemaical funcion which depends on Where, ime, wha implies hen by properies of he mean ha in each year he observaions of he populaion will be PP ii, Denoes he slope of he sraigh-line beween (PP ii, ii ) and (PP ii+, ii+ ) of he wo dimensional space Global Journals

5 defined by ime and he populaion (Leihold, 973, p. 37), and pp MMMM ii Represens he middle value beween he populaion daa denoed as PP ii and PP ii+ The able has he resuls of he calculaions and in figure in X axis are middle values of he populaion, and he slope values are in Y axis S l o p e s g(p) Figure : The slopes and middle poins of he populaion in México, As we can see in figure he behavior of slope in erms of he middle values is also random, so ha, according o he probabiliy heory mus also have a mean and a variance, and as a consequence of he second posulae of he Sable Bounded Theory is mean mus be equal o oher unnown mahemaical funcion ha we will denoe wih he leer g. I is imporan also o poin ou ha funcion g depends of populaion. In figure, we can also observe ha funcion g seems o be a parabola, so ha, if his assumpion Middle values Source: Table is rue mus here be wo values of he populaion where he g funcion s curve inersec he X axis. Those values we will denoe hem as KK and KK + KK. Bu besides, is imporan poin ou ha in hose values he change amoun regard ime is zero, wha implies KK + KK is a maximum value for he populaion and KK is a minimum value. This fac proves empirically ha mean of Mexico s populaion is bounded by hose values. Table : Time, populaion, middle poins and slopes in Mexico Year Time Populaion Middle Poins Slopes Global Journals

6 Source: Table and own calculaions based on equaions 3 y 4; Time was calculaed as Year-790 To proving mahemaically exisence of he maximum and minimum and besides o finding esimaors of hem, we adjus a regression model o he daa of figure, his is, values of he populaion ha mae he slope of deerminisic componen of 5 is zero, ha is, 0 = AP + BP + C i i P = AP i + BP i + C + ω (5) i and afer, using he formulas o calculaing he roos of i a parabola we have ha P µ = AP + BP C i + (6) i B B 4 AC Where, = + (7) A A PP, Denoes he slope, PP ii, Denoes he populaion, B B 4 AC + = (8) A A AA, BB aaaaaa CC, are unnown consans, ωω ii, are random variables ha we suppose independens, These resuls indicae ha formulas 7 and 8 are wih disribuion law Normal, mean μμ ωω = 0, and consan esimaors of he minimum and maximum values of he variance σσ ωω, and populaion respecively. The following able 3 presens PP μμ, Denoes he mean of populaion. he esimaes of leas squares ordinary of he consans A, B and C, and he p-values o deermining heir From he mahemaical poin of view, he saisical significance. maximum and minimum values are equal o hose Table 3: Parameers esimaed of he equaion 6 and p-values o prove is saisical significance Parameer Esimaion Esandar Error -Value p-value A B C Source: Own calculaions based on he middle poins and slopes of able and Saa/SE. As we can seen, he hree coefficiens are significanly differen from zero, so ha, o esimae he maximum and minimum values of he populaion, he = + ( ) = ( ) esimaions of he coefficiens were subsiued in 7 and 8, obaining ha, = = In addiion o he significance of he parameers, value of he F Saisic was.87 wih a p-value of , which proves ha he parabola assumpion in 6 is rue wih a deerminaion coefficien 90.9%. These resuls ogeher wih he fulfillmen of he The residual analysis indicaes ha he random variables of he model are disribued normal, are independen and have consan variance. 4( ) ( 0.357) ( ) 4( ) ( 0.357) ( ) 53.6 assumpions of he residuals of 5 prove mahemaically he exisence of he maximum and minimum of he Mexican populaion. Finally, i is imporan o clarify ha he values KK = 7. and KK + KK = 53.6 are bounds for he mean of he populaion, bu no for he observaions, which according o he probabiliy heory hey will deviae a cerain amoun around he mean depending on is 5 08 Global Journals

7 6 variance, herefore hey can be greaer or lesser han KK = 7. and KK + KK = 53.6, bu heir occurrence will be governed by a probabilisic law. d) The paern of populaion growh in Mexico According o he posulaes of he Sable Bounded Theory, he behavior equaions of he observaions and mean of he populaion in each ime are, P = f ( ) + ε µ P = f () The problem is ha in pracice he funcion ff() is unnown, however, he rend of daa and he exisence of he maximum and minimum values can give us idea of how is is derivaive, and he heory of differenial equaions can help us o deducing is mahemaical equaion. Firsly, according o rend of observed daa, he funcion ff() has o be increasing, and so, is derivaive will be posiive. Secondly, due o exisence of he maximum and minimum values is derivaive will have o be zero in hose values. Based on hese properies he Sable Bounded Theory deduces a funcion which saisfies hose properies menioned. The Sable Bounded Theory begin supposing ha derivaive of he unnown funcion is given by he produc of wo funcions h (PP) and h (), one ha depends of he populaion and oher ha depends of ime, forming a differenial equaion of separable variables (Wilye, 979), which has as soluion a funcion ha relae he populaion and ime, namely, Where dp = h ( P) h ( ) d (9) DP, denoes he derivaive of ff() d Now since he derivaive mus be posiive and equal o zero in he minimum and maximum values, hen he funcion h (PP) can be as follows: And so, dp d h ( P) = ( P ) ( P ) = ( P ) ( P ) h ( ) Where h (PP) KK and KK + KK are he minimum and maximum values. We can observe ha due o KK and KK + KK are bounds inferior and superior respecively of he populaion, hen quaniy (PP KK ) is always posiive, bu quaniy (PP KK KK ) is negaive, herefore (PP KK ) (PP KK KK ) is negaive, wha implies h () mus be negaive in order o he derivaive be posiive as we require. By oher hand, we can also see ha when he populaion is equal o KK and KK + KK and hen he derivaive is zero, he oher condiion we require. Now separaing variables we have dp = ( P )( P ) h ( ) d Solving by parial fracions he indefinie inegral of he lef hand we have ha P = + (0) λ ( ) + e Where λλ() is an unnown funcion such ha is derivaive is equal o h () and which can be deermined by using he observed daa, since, Ln P = λ( ) () Wha implies if he Sable Bounded Theory is KK rue ha variable mus be a funcion of ime. PP KK Gonzalez - Rosas (00) call o his variable he ransformed of he populaion. e) Esimaion of he λλ() funcion In order o esimae he funcion λλ() firs we esimaed KK and afer subsiue esimaions on equaion, his is, = 7. + = 53.6 =46.5 Afer ha, we assign he values observed of he populaion and calculaed he ransformed of populaion. In able 4 we can see resuls and in figure 4 he behavior of he ransformed and ime. 08 Global Journals

8 Table 4: Year, ime, populaion and ransformed of he populaion in Mexico, Year Time Populaion Transformed of he Populaion Source: Own calculaions based on equaion. The ransformed of he years 790, 803, 80, 80, 838 and 84 were declared no defined because does no exis he naural logarihm of a negaive number. The ransformed of year 850 was no considered because was an oulier. Tranformed of he populaion T i m e Source: Table 4 Figure 3: The ransformed of he populaion and ime in Mexico, As we can see in figure 3 he ransformed of he populaion and ime are relaed by a parabola, his is, λλ() = AA + BBBB + CC. However, we can also observe a sraigh-line paern afer ime 50, wha would imply λλ() = αα + ββββ. In figure 4 we can observe he relaion. So, in order o compare he wo paerns we adjused boh funcions o he observed daa Global Journals

9 T i m e Source: Table 4 Figure 4: Transformed of he populaion and ime in Mexico, Bu, due o he derivaive of λλ() has o be equal o h () which has o be negaive, so, in he case of sraigh-line paern he β parameer has o be negaive, and in he case of parabola paern he parameers A and B have o be negaives. If we use he sraigh-line we obain a paern called Logisic, bu if we use he parabola we have a paern called Exended Gaussian. The equaions of hese paerns are respecively, P = + () α + β + e P Transformed of he populaion (3) + e = + A + B+ C In he equaion, due o β is negaive when ime is increased hen is P is near o KK + KK, and in equaion 3, because A and B are negaives when ime is increased P is near also o KK + KK. Tha is, hose parameers deermine how quicly P approaches he maximum. Due o hese characerisics he parameers β, A and B are called he quicness parameers (González-Rosas, 08). In order o deermining wha paern is adjused beer o observed daa we esimaed boh he sraighline and he parabola. The following able presens he ordinary leas squares esimaion and he p-values of he sraigh-line and parabola. Table 5: Parameers esimaed of he equaion and p-values o proving is saisical significance Parameer Esimaion Sandar p-value Error of es α β A B C As we can see in able 5 all parameers are significanly differen of zero wih a 5% significan level, excep he B parameer which is significan a 6% level. We can also see he p-values of boh F ess ha indicae boh equaions are correc a 5% significan level. Finally, we have he deerminaion coefficien which poin ou ha sraigh line explain he percen of he daa variaion of ransformed populaion, while parabola explain 99.7 percen, his is, he sraigh - line explain daa variaion beer. And so, esimaed equaions of he logisic and Gaussian paerns are respecively, p-value of F Tes R Source: Own calculaions based on able 4 and Saa/SE. P 46.5 = 7.+ (4) e P = 7.+ (5) e In figure 5, we have he graphics of boh paerns. As we can see a glance boh logisic paern and Gaussian paern fi very well o he observed daa a all period, however, he Gaussian paern seems o is adjused beer han Logisic paern in period. Bu by oher hand, Logisic paern seems o 08 Global Journals

10 adjus beer han he Gaussian one in and periods. However hese crieria are very ambiguous, so, we had o define a Measure of he Adjus Error as i follow: 05 = 9 MAE = ( y y ) (6) Where, MAE is he measure of he adjus error, yy Denoes he observed populaion in year, and y is he esimaed populaion in year Populaion Source: Table and own calculaions based on 4 and 5 Figure 5: Observed daa and logisic and Gaussian paerns in Mexico, When we subsiued daa a equaion 6 we found ha he MAE for he logisic paern was 98.73, while for he Gaussian paern was 0.6. Based on hese resuls we decided ha Logisic paern explain beer he populaion evoluion hrough ime in Mexico. IV Resuls a) Puncual forecass of he Populaion in Mexico All he resuls above prove ha behavior of he mean of he populaion hrough ime in Mexico is governed by following mahemaical equaion: P 46.5 = 7.+ (7) e 0474 Where, PP, Denoes he mean of he populaion in ime, The consan 7. is he minimum value of Mexico s populaion, The consan =53.6 denoes he maximum value of Mexico s populaion, and The consan is he quicness parameer of Mexico s populaion. So ha, when we gave values o ime variable in equaion 7 we obained puncual forecass of he mean of Mexico s populaion for he period (Annex ). In figure 6, we can observe ha he model is adjused very well o he observed daa. Observed Logisic Gaussian 9 08 Global Journals

11 0 Populaion Observed Forecas Source: Table and annex Figure 6: Observed and forecased populaion in Mexico, According o he resuls of equaion 7, we found ha in 05 he mean of Mexico s populaion will be 30. million of people, in 040 will be 4., and for 050 he mean of Mexico s populaion will reach 48 million of people. Sill 8 million per under he populaion maximum ha is of 53.6 people. I is very imporan o clarify ha wha we are forecasing is he populaion mean no he observaions, because, hose are random and hence canno be predicable, so ha, in 05 he real observaion can be below or above o he 30. million, he same will happen in 040 and 050. V. Discussion If we consider ha in each momen of ime he populaion is a random phenomenon, hen we can explain behavior irregular observed of he populaion in figure, however, his hypohesis brough as a consequence ha we canno forecas he populaion, since random phenomena canno be prediced. So, he quesion arose, how can we predic wha is no predicable? The answer arrived us of he probabiliy heory, since, according his heory if he populaion is a random phenomenon mus o have a mean and a variance, so ha, when we supposed ha mean had a deerminisic behavior given by a mahemaical funcion ha depends of ime, hen we accep ha we would be able predic o leas he mean of he populaion. Afer ha, according o rend of daa, he funcion had o be growing, however populaion canno grow, grow and grow, so ha, was beer suppose ha mus end o he sabilizing or maybe o reach a maximum and afer ha, decrease. This siuaion brough us wo quesions more, firsly, wha is he value, where he populaion is going o sabilize or reach he maximum in fuure? And secondly, wha is he funcion we had o use o predic he populaion? This wo quesions we answered hem using he Sable Bounded Theory, which allowed us o prove exisence of a sabilizer value and besides o calculae i. Also we find he funcion or paern which allowed us o do he predicions of he populaion. VI. Conclusions In Mexico, for he period , he behavior of mean of populaion hrough ime is governed by a mahemaical low ha depends of ime. By firs ime, he scienific communiy has mahemaics ess abou subjecs ha we only wach a glance, ha is, ess abou he logisic paern of he populaion growh. In Mexico in order o explaining evoluion of he populaion rough ime, he demographers have used he logisic paern, however, never hey have given a mahemaic es, his paper prove all he hypohesis used abou i and subsiue he empirical aspecs. Alhough his exercise was done wih daa from Mexico, i is imporan o mae i clear ha he Sable Bounded Theory can be applied o any counry where daa on he populaion are available. Also i can be used o forecasing moraliy, feriliy and ne migraion. However, i is necessary o warn ha he resuls of his paper are based on he assumpion of he social, economic and poliical condiions of Mexico will coninue wihou change. If his assumpion i is no fulfill, he forecass we are giving will no be rue. Also i is necessary o warn ha he mahemaical modeling of realiy is based on assumpions, and ha heoreical resuls are rue only if he assumpions fulfill, so ha, i is necessary o do a grea effor o prove ha he assumpions are rue. Finally, is clear ha any exercise o predic he fuure is exposed a lo of error sources: wrong daa, false assumpions, and wrong models, so on. Therefore, i is necessary o idenify all possible error sources, and hen uilize mehodologies ha minimize hose errors. The Sable Bounded Theory is an example of ha. 08 Global Journals

12 References Références Referencias. Alema, L., Gerland, P., Rafery. A. & Wilmoh, J. (05). The Unied Naions Probabilisic Projecions: An inroducion o demographic forecasing wih uncerainly. Foresigh (Colcheser, V), 05(37), 9-4. Recovered on April 0, 07 from hps:// Ausralian Bureau of Saisics Saisical Languaje. Esimae and Projecion (07), Recovered on June 4, 07 from hp:// a30.nsf/home/saisical+language+-+ esimae+and+projecion 3. INEGI (07a), Sisema para la Consula de las Esadísicas Hisóricas de México 04. ["Sysem for he Consulaion of Hisorical Saisics of Mexico 04"]. Recovered on May, 07 from: hp://dg cnesyp.inegi.org.mx/cgi-in/ehm04.exe/ci INEGI (07b), Principales resulados de la Encuesa Inercensal de 05. ["Main resuls of he Inercensal Survey of 05"]. Recovered on April 8, 07 from hp:// proyecos/enchogares/especiales/inercensal/05/ doc/eic05_resulados.pdf [NEGI (07). [Main resuls of he 05 Inercensal Survey] 5. Gonzalez-Rosas, J. (00), Teoría Esadísica y Probabilísica de los Fenómenos Esable Acoados, Tesis de maesría. Universidad Nacional Auónoma de México. [Saisical and Probabilisic Theory of Phenomena Sable - Bounded, Maser hesis, Naional Universiy Auonomous of Mexico]. 6. González - Rosas, J. (0), La Teoría Esable Acoada: Fundamenos, concepos y méodos, para proyecar los fenómenos que no pueden crecer o decrecer indefinidamene. Saarbrucen, Alemania. Ediorial Académica Española. [The Sable Bounded Theory: Fundamenals, conceps and mehods, o projec phenomena ha canno grow or decrease indefiniely. Saarbrucen, Germany. Spanish Academic ediorial]. 7. González-Rosas, J. (08), The Sable Bounded Theory. An alernaive o projecing he ne migraion. The case of México. In Ahens Journal of Social Sciences, volume 5, Issue, January 08. Ahens, Greece. 8. Leihold, L. (973), El Cálculo: Con geomería analíica (a Edición), México, Harla S.A. de C.V. [The calculaion wih analyic geomery. ( nd Ediion), Mexico, Harla S.A. Of C.V]. 9. Medhi, J. (98), Sochasic Processes, ( nd Ediion), New Yor, John Wiley & Sons. 0. Parida B. V. (006). Proyecciones de la Población de México , CONAPO, México. [Projecions of he Populaion of Mexico CONAPO. Mexico].. Populaion Reference Bureau (07), Undersanding and using Populaion projecions. Recovered on June 4, 07 from p:// Repors/00/UndersandingandUsingPopulaionPr ojecions.aspx. The Cohor Componen Mehod for Maing Populaion Projecions (07), Recovered on June 4, 07 from hp:// echcoop/popproj/module/chaper.pdf 3. Unied Naions. Deparmen of Economic and Social Affairs (07), Recovered on June 4, 07 from hp:// laion/05-repor.hml 4. Wilye C. Ray (979), Differenial equaions, McGraw Hill. Mexico, pp World Populaion Hisory (07), Projecing Global Populaion o 050 and Beyond, Recovered on June 4, 07, from hp://worldpopulaionhisory. org/projecing-global-populaion/ Global Journal of Managemen and Business Research Volume XVIII Issue III Version I Year 08 ( G ) 08 Global Journals

13 Annex Populaion Forecass in Mexico, Year Time Forecas Year Time Forecas Source: Own calculaions based on equaion 7 08 Global Journals