The Stable Bounded Theory: A Solution to Projecting the Total Fertility Rate in Mexico

Transcription

1 Ahens Journal of Socal Scences XY The Sable Bounded Theory: A Soluon o Projecng he Toal Ferly Rae n Mexco By Javer González-Rosas Ilana Zárae-Guérrez Toal ferly rae (TFR) s one of he demographc componens ha deermne changes n populaon volume; hus, f we requre populaon esmaes n he fuure, s absoluely necessary o projec hs componen. However, projecon of he TFR has wo nheren problems: frs, we have o deermne ha value where he TFR wll sablze n fuure, whch s called he sablzer value, and second, we have o deermne he funcon whch we are gong o use o projecng he TFR. The sable bounded heory provdes us soluons o hese wo nheren problems wh sold scenfc suppor. The frs problem s solved by fndng an esmae of he sablzng value ha depends on he observed daa of he ndcaor. The second s solved by deducng a funcon ha s sablzed precsely a he esmaed sablzer value. The man resuls of he arcle ndcae ha he TGF of Mexco wll sablze a 1.98 chldren per woman and ha he funcon, alhough belongs o he exponenal famly, s no a logsc funcon, whch has radonally been used o projec he TFR. Oher resuls ndcae ha he average TFR n Mexco n 2020 wll be 2.16 chldren per woman; n 2030 wll be 2.07; n 2040 wll reach he fgure of 2.03; and by 2050, wll be 2.0 chldren per woman. Keywords: Forecas, Ferly Specfc, Sably, Toal ferly rae. Inroducon Populaon projecons n Mexco usually have been elaboraed by applyng he mehod of demographc componens, whch s based on esmaons of brhs, deahs, emgrans and mmgrans, whch effec changes n human populaons. Ths mehod esmaes brhs and deahs by projecng ferly and moraly usng logsc funcons. However, emgrans and mmgrans have no been projeced usng mahemacal models; raher, her projecon s resrced only o assumpons abou her fuure behavor. The demographc componens mehod provdes nformaon besdes he populaon volume, such as he demographc dynamcs of he counry by predcng he fuure behavor of componens lke ferly, moraly and nernaonal mgraon. In Mexco, fuure ferly s projeced frs by he level and second by he srucure. The level s obaned by projecng he TFR, whch radonally s esmaed by adjusng a logsc funcon o he observed daa. The use of he logsc funcon s jusfed because n Mexco he observed daa of ferly hrough me follows a logsc paern and s sablzed n he Drecor of Soco economc Sudes and Inernaonal Mgraon, Naonal Populaon Councl, Mexco. Depuy Drecor, Secreara of he Ineror, Mexco. 1

2 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... fuure. The problem s ha here are a leas wo oher funcons ha can also be adjused o he observed daa accepably and ha also sablze n he fuure. Nowadays, he value where ferly wll sablze s no esmaed, bu s fxed by an expers group convened by he Uned Naons. An age srucure projecon s done, frs by defnng he lm srucure as he age composon of Uned Naons (UN) projecons ha correspond o he TFR fxed by he expers, and second by dong lnear nerpolaons beween lm srucure and observed age composon (Parda-Bush 2008). The Sable Bounded Theory solves he wo nheren problems ha are presen when we wan o forecas ferly, n oher words he sably value problem and he problem of he funcon used for s projecng. Leraure Revew Accordng o he daa of he 2015 Revson of World Populaon Prospecs, he TFR s now 2.5 chldren per woman globally. However, hs global average masks wde regonal dfferences. Afrca connues beng he regon wh he hghes ferly, snce s TFR s 4.7 chldren per woman, whle Europe has he lowes ferly, wh a TFR of 1.6 chldren per woman. Oher regons, such as Asa, Lan Amerca and he Carbbean, have a TFR of 2.2, closely followed by Oceana wh 2.4 chldren per woman (Uned Naons 2015). Ferly declne has been an mporan deermnan of populaon agng. Consequenly, ferly projecng has mporan mplcaons for he age srucure of fuure populaons and he pace of populaon agng. The TFR s one of he key componens n hese populaon projecons. Whn a specfc perod, s defned as he average number of chldren a woman would have f she survved hrough he end of he reproducve age, experencng a each age he age-specfc ferly raes of ha perod (Alkema e al. 2011). In 2015, close o half of he world s populaon lved n counres where he perod oal ferly raes (TFR) were below he replacemen level 1. Ths ncluded many mddle- and lower-ncome counres n Asa and Lan Amerca, such as Brazl, Chna, Iran, Thaland, Turkey and Venam. Some counres of Europe and Eas Asa even experenced levels of "lowes-low ferly" wh a TFR fallng below 1.3 chldren per woman. The curren low ferly s somemes vewed as emporary, caused by posponemen of chldbearng, economc uncerany, a passng phase n he process of human developmen, or n he ongong "gender revoluon". In order o face he effecs of hese low ferly levels, governmens wll have o desgn new publc polces (Soboka e al. 2015). In he UN, he populaon s projeced usng a cohor componen mehod. Accordng o hs mehod, s necessary o observe a populaon n a base year, as well as he fuure ferly raes, survval probables and mgraon ne couns. All of hese measures are aken n fve-year age groups and accordng o gender. Gven hose npus, he mehod assesses he populaon by boh age 1 The replacemen level means ha a woman wll have a daugher alve when her reproducve perod fnshes, wha means, he woman wll be replace o connung he reproducve process. 2

3 Ahens Journal of Socal Scences XY and sex n year +5, addng o he populaon n year he brhs and ne mgraon occurrng durng a 5-year perod, mnus he deahs ha also occurred durng he same perod. Tradonally, he UN has produced deermnsc populaon projecons and puncual (ha s, only one number). These pon projecons were supplemened wh ranges based on dfferen scenaros of demographc changes. In July 2014, he UN for he frs me ssued offcal probablsc populaon projecons, whch quanfy he uncerany assocaed wh demographc projecons (Alkema e al. 2015). An old queson n economcs ha goes back a leas unl he me of Malhus s: how much hgher should ncome per capa be f he ferly rae were o fall by a specfed amoun? Over he las half cenury, he consensus abou he effecs of ferly declne have changed from seeng hem as srong o no very mporan, and recenly back oward assgnng hem some sgnfcance. For an ssue ha has been suded for so long, and wh such poenal mpor, he evdence regardng he economc effecs of ferly s raher weak. However, n 2013 Ashraf found ha a reducon n ferly rases ncome per capa by an amoun ha some would consder economcally sgnfcan. Alhough he effec s small relave o he vas gaps n ncome beween developed and developng counres, he fndng s a poenal answer o he old queson (Ashraf e al. 2013). Europe has already compleed s demographc ranson. Demographc ranson heory has been very useful for explanng global demographc rends durng he 20h cenury, and sll has srong predcve power when comes o projecng fuure rends n counres wh hgh ferly. However, s no useful for predcng he fuure of ferly n Europe nowadays. The curren noon of a second demographc ranson s a useful way o descrbe a bundle of behavoral and normave changes ha recenly happened n Europe, bu has no predcve power. The problem s ha here s no ye a useful heory o predc he fuure ferly level of pos-demographc ranson socees; we do no even know wheher he rend wll go up or down (Luz 2006). Mehodology Daa Used The sources of he daa used n hs paper are nne surveys wh naonal represenaveness regardng ferly and ha have been conduced n Mexco durng he perod Table 1 presens he specfc and oal ferly raes used n hs arcle. The 1975 daa were esmaed by he Secreara of Programmng and Budge hrough he General Drecorae of Sascs based on he Mexcan Ferly Survey (MFS) of The daa for 1978 were esmaed by he Coordnaon of he Naonal Famly Plannng Program based on he Naonal Survey of Prevalence n he Use of Conracepve Mehods (NSPUCM) of The daa of 1991, 1996 and 2005 were he auhors own calculaons based on he Naonal Surveys of he Demographc Dynamcs (NSDD) of 1992, 1997 and The daa of 2002 were also unque calculaons based on he 3

4 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... Naonal Reproducve Healh Survey (NRHS) of Fnally, he daa for 2009 and 2005 were esmaed by he Naonal Insue of Sascs and Geography (INEGI by s acronym n Spansh). Table 1. Ferly Specfc Rae per Age Group n Mexco, Year TFR Source: 1975) Secrearía de Programacón y Presupueso, 1979, p. 138; 1978) Coordnacón del Programa Naconal de Planfcacón Famlar, 1979, p. 27; 1986) Dreccón General de Planfcacón Famlar, 1989, p. 36; 1991, 1996, 2005) Own calculaons based on he NSDD of 1992, 1997 and ) Own calculaons based on he NRHS of 2003; 2009, 2014) INEGI, prncpales resulados de la Encuesa Inercensal de 2015 Srucure and Level of he TFR n Mexco a Las 40 Years In Mexco he avalable daa ndcae ha he ferly srucure has undergone very mporan changes durng he perod In 1978, accordng o he NSPUCM, ferly srucure was characerzed by a curve wh a lae peak, snce he greaes ferly was n he group years old. In 1991, accordng o he 1992 NSDD, he ferly curve changed o an earler peak, due o he greaes ferly occurrng n he group years old. For 2014, ICS s daa of 2015 show ha he ferly curve has an exended form, snce he ferly raes n groups and years old are very smlar (Fgure 1). Fgure 1 also shows ha he groups ha conrbue he mos o he level of ferly (called majory groups) n all years are 20-24, 25-29, and years. The conrbuon of hese groups o he level of ferly has no changed sgnfcanly. In 1978, he conrbuon of hese groups was 83.6%, n 1991 of 82.5% and n 2014 of 83.0%. On he oher hand, he groups ha have a smaller conrbuon (called mnory groups) are 15-19, and years. These groups accouned for 16.4%, 17.5% and 17.1% n he years 1978, 1991 and 2014, respecvely. 4

5 Chldren per woman Chldren per woman Ahens Journal of Socal Scences XY Fgure 1. Ferly Specfc Rae per Age Group n Mexco, 1978, 1991 and Source: Table Bu a reducon of ferly specfc raes n he counry mpled ha he TFR also fell drascally n he las 40 years. In 1975, accordng o MFS, he TFR was esmaed a 6.03 chldren per woman, whle n 1991 and accordng o he 1922 NSDD, was esmaed a 3.5 chldren per woman, whch represened a reducon of almos hree chldren n 17 years. However, snce hen he speed of reducon of TFR n Mexco has decreased. In he las 23 years only reduced by abou one chld, from 3.5 n 1991 o 2.29 chldren per woman n 2014, accordng o he ICS of 2015 (see Fgure 2). Fgure 2. Toal Ferly Rae n Mexco, Source:Table 1 Toal Ferly Rae Sably A frs glance, he mos recen observaons of he las 40 years of he TFR ndcae ha has sopped descendng wh he velocy ha was observed earler 5

6 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... (see Fgure 2). Thus, f he TFR ends o sablze hen s change velocy mus be close o zero; herefore, he TFR mus no change over me, namely, mus be equal o a consan. Ths consan s called TFR s sablzer value, and he answers regardng s exsence and he calculaon of hs consan seem o be n he Sable Bounded Theory (Gonzalez-Rosas 2012). The Sable Bounded Theory ress n wo fundamenal posulaes. Frs, n each year he TFR s a random phenomenon, and so accordng o probably heory, n each year mus have a mean and a varance. Second, he mean of he TFR s equal o a mahemacal funcon, whch depends on me, and hus mples by properes of he mean ha n each year observaons of he TFR wll be equal o a quany deermned by he mahemacal funcon plus a ceran random devaon, whch occurs accordng o probablsc law. Medh (1981) called he mahemacal funcon he deermnsc componen and he random devaon s called he sochasc componen. Under hese posulaes hen, he behavor equaons of observaons and he mean of he TFR n each me would be: Where: f ( ) (1) f () (2), denoes he observaon of he random varable of he TFR n me, f (), s a mahemacal funcon unknown,, are random varables ha we suppose ndependens, wh dsrbuon law 2 Normal, mean 0, consan varance, and, denoes he random varable mean TFR n me. Due o he change velocy of he TFR beween a me and oher 1 s measured wh he slope of he sragh lne ha jons wo pons of a bdmensonal space defned by me and he TFR. In order o es he sably hypohess, we calculaed slopes and mddle values 2 of wo consecuve TFR values of he followng form: 1 VM (3) (4) 2 The Sable Bounded Theory proves ha exs hree esmaors of sablzer value. One of hem s assocaed wh y, oher wh y+1, and one more wh mddle value beween he wo. Ths heory also proves ha he bes of he hree s mddle value. 6

7 Ahens Journal of Socal Scences XY Where:, denoes slope of sragh lne beween (, ) and ( 1, 1) of wo dmensonal space defned by me and he TFR (Lehold, 1973, p. 137), and VM, represens mddle value beween TFR daa denoed as and 1. In Table 2, you can see he resuls of calculaons and n Fgure 3 mddle values of he TFR are on axs X, and slope values are on axs Y. Table 2. Toal Ferly Rae, Mddle Pons and Slopes n Mexco Year Tme TFR Mddle Pons Slopes Source: Table 1 and own calculaons based on equaons 3 and 4. As can be seen n Fgure 3, he pons are no exacly on he sragh lne, wha can be explaned f we suppose ha slope s also random, and so, under hs hypohess accordng o he probably heory he slope mus have a mean and a varance, wha would mply ha he pons would be he observaons of he slope whle he sragh lne would be s mean, mahemacally hs suaon would be able represened as: Where: (5). (6), denoes he observaon of he random varable of he slope,, denoes he TFR, and, are unknown consans, and, are random varables ha we suppose ndependens, wh dsrbuon law Normal, mean 0, consan varance. 2 7

8 Slopes Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... Fgure 3. Slopes and Mddle Pons of he Toal Ferly Rae n Méxco, Source: Table Mddle pons of he TFR You can also see n Fgure 3, ha when mddle values of he TFR are reduced, he pons approach zero and also he slope values approach zero. Furhermore, you can also observe ha he sragh-lne nersecs axs X near he 2 value, whch emprcally proves ha he sablzer value of he TFR exss. From a mahemacal pon of vew, he sablzng value s equal o he value of he TGF ha makes he slope of Equaon 6 become zero, ha s: 0 and hen, f we do some algebrac operaons, we fnd ha he sablzer value of he TFR denoed as K s: K (7) Ths resul ndcaes ha o calculae he sablzer value of he TFR, s necessary o calculae he consans and of Equaon 6. To esmae hese consans, a smple lnear regresson model was fed o he daa of Fgure 3. The followng able presens he ordnary leas squares esmaes of he and parameers and he p-values o deermne her sascal sgnfcance. Table 3. Parameers Esmae of he Equaon 6 and p-values o prove s Sascal Sgnfcance Parameer Esmae Sandard error -value p-value Source: Own calculaons based on mddle pons and slopes of able 2. 8

9 Ahens Journal of Socal Scences XY As can be seen, he wo coeffcens are sgnfcanly dfferen from zero, so ha o esmae he sablzer value of he TFR, he esmaons of he coeffcens and were subsued n 7, hus obanng he concluson ha TFR n Mexco wll sablze wh he value: K K 1.98 In addon o he sgnfcance of he parameers, he p-value of he F sasc was , whch proves ha he sragh-lne assumpon n 6 s rue, and he coeffcen of deermnaon was 76.39% 3. These resuls prove mahemacally he exsence of he sably of he TFR. A parabolc model also was fed o he same daa, bu he coeffcen of he quadrac erm was no sgnfcan, whch proves hen here are no wo sablzng values for he TFR of Mexco. Fnally, s mporan o clarfy ha he value K = 1.98 s a bound for he mean of he TGF bu no for he observaons, whch accordng o he heory of probably wll devae a ceran amoun around he mean dependng on s varance. Therefore hey can be greaer or less han K =1.98, bu her occurrence wll be governed by a probablsc law. The Equaon of he TFR and Tme Accordng o he posulaes of he Sable Bounded Theory, he behavor equaons of he observaons and mean of he TFR n each me are: f ( ) f () The problem s ha n pracce he sablzer funcon f() s unknown. However, he rend of he daa and he exsence of he sablzer value can gve us an dea of wha s dervave s, and he heory of dfferenal equaons can help us o deduce s mahemacal equaon. Frsly, accordng o rends of observed daa, he funcon has o be decreasng, and so, s dervave wll be negave. Secondly, due o he exsence of a sablzer value, s dervave wll have o be zero n he sablzer value. Based on hese properes, he Sable Bounded Theory deduces a funcon ha sasfes he properes menoned. The Sable Bounded Theory supposes ha he dervave of he unknown funcon s gven by a produc of wo funcons, h 1( ) and h ( ) 2, one ha depends on he TFR and oher ha depends on me. These form a dfferenal 3 The resdual analyss ndcaes ha he random varables of he model are dsrbued normal, are ndependen and have consan varance 9

10 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... equaon of separable varables (Wlye 1979), whch has as a soluon a funcon relang he TFR and me, namely: d h1 ( ) h2 ( ), (8) d Now, snce he dervave mus be negave and equal o zero n he sablzer value K, he funcons h ( ) and h ( ) can be as follows: h1 ( ) ( K) h ( ) m d d ( K) m Where m s a consan less han zero and K s he sablzer value. We can observe ha due o K beng an nferor bound of he TFR, hen quany ( K) s always posve, bu when you mulply by m, hen ( K) m s negave. Ths dervave s negave as we requre. On he oher hand, when he TFR s equal o K hen he dervave s zero, anoher condon we requre. Now separang varables we have: 1 d ( K) m d Solvng he ndefne negrals, we arrve a: ln( K) m C Fnally, clearng he varable we oban: m K e ; con m 0 (10) Where denoes he TFR a me, K he sablzer value, e s he exponenal funcon, and and m are unknown consans, such ha defnes he nal condons of he TFR a me zero and m represens he amoun of reducon per un me. Snce he parameers and m deermne how quckly he TFR approaches sably, hey are called he parameers of speed. We can observe ha snce m s negave, Equaon 10 s decreasng; furher, m when ends nfne, e ends zero, so ha he TFR ends owards K, he sablzer value. Equaon 10 s called he decreasng exponenal funcon and 10

11 Ahens Journal of Socal Scences XY solves heorecally he problem of exsence of a sablzer funcon of behavor equaons, boh of observaons as he mean of he TFR. Esmaon of he Parameers of he Speed Accordng o Draper and Smh (1966), Equaon 10 s no lnear a parameers, m and K, so hey canno be esmaed by he mehod of leas squares. However, f a Equaon 10 we pass K o he lef of equaly and f we apply he naural logarhm a boh sdes of he equaon, we oban: K ln m ln (11) Tha s, he resul s a lnear equaon a he parameers ln and m, whch can be esmaed by he mehod of ordnary leas squares or generalzed leas squares. Ths suggess ha he esmaon of he parameers of Equaon 10, can be done n wo sages. Frs, we esmae K and hen we esmae ln and m 4. ln K s called he ransformed of he TFR. The varable Table 4. Year, Tme, TFR and Transformed of he TFR n Mexco Year Tme TFR Transformed of he TFR Source: Tme was calculaed as year-1975; Transformed was calculaed based on equaon 11. Replacng he value K=1.98 n Equaon 11, we calculaed he ransformed of he TFR (see Table 4). In Fgure 4 we can check ha he relaon beween he ransformed of he TFR and me n Mexco s gven effecvely by a sragh lne, as s predced by he Sable Bounded Theory. Thus, s lnear a parameers ln and m. In order o esmae hem, we adjused a smple regresson model o daa of Fgure 4. 4 The resdual analyss ndcaes ha he random varables of he model are dsrbued normal, are ndependen and have consan varance. 11

12 Transformed Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... Fgure 4. Transformed of he TFR and me n Mexco, Source: Table 4 Tme Table 5 presens he ordnary leas squares esmaes of he parameers and he p-values ha prove her sascal sgnfcance. Noe ha boh parameers are sascally sgnfcan wh values ln and m To oban he esmaon of he, we apply an exponenal funcon o obanng Table 5. Parameers Esmaon of he Equaon 11 and p-values o prove s Sascal Sgnfcance Parameer Esmaon Sandard error -value p-value ln M Source: own calculaons based on able 4. Wh he esmaon of he parameers of speed, he equaons of behavor of he observaons and of he average of he TFR as a funcon of me were compleely solved. The equaon of he observaons can be used o elaborae nerval projecons of he TFR, whereas he equaon of he mean can be used o carry ou pon projecons of he mean of he TFR. Resuls Puncual Forecass of he TFR n Mexco The resuls above prove ha behavor of he mean of he TFR hrough me s governed by followng mahemacal equaon: e 1.98 (12) 12

13 Chldren per woman Ahens Journal of Socal Scences XY Where, denoes he mean of he TFR a me, The consans and deermne he nal condons of he mean of he TFR a me zero and he amoun of reducon of he mean of TFR per un me, respecvely. The consan 1.98 represens he sablzer value of he mean of TFR. Fgure 5. Toal Ferly Rae Observed and Esmaed n Mexco, Observed Esmaed Source: Table 1 annex Gvng values o he me varable n Equaon 12, we obaned puncual forecass of he mean of he TFR n Mexco for he perod In Fgure 5, you can observe ha he model s adjused very well o he observed daa, and when me s ncreased he mean of he TFR approaches he sablzer value. Accordng o he resuls of Model 12, we found ha n 2020 he mean of TFR wll be 2.16 chldren per woman; n 2030 wll be 2.07; n 2040 we expec ha wll be 2.03; and fnally, n 2050 wll have reached sably arrvng a 2.0 chldren per woman. Dscusson When we analyze TFR s daa hrough me we realzed us ha hey had a rregular behavor bu wh a decreasng rend, such ha, when we consder ha n each momen of me he TFR rae s a random phenomenon, hen we could explan behavor rregular observed of he rae. However, hs hypohess mpled ha we could no forecas he oal ferly rae, snce random phenomena canno be predced. Bu f he objecve of our paper was predc he TFR, so he queson arose, how can we predc somehng ha s unpredcable. The answer comes ou of probably heory. Accordng o hs heory, he oal ferly rae mus have a mean and a varance, so ha, when we assumed he mean had a deermnsc behavor gven by a mahemacal funcon dependen on me, hen we would be able predc a leas he mean of he oal ferly rae. Accordng o he rend of daa, he funcon had o be decreasng, 13

14 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... however ferly canno ge down unl arrves o zero, snce evdence of some developed counres ndcae ha s gong o sablze, hs suaon brough us wo more quesons. Frsly, wha s he value where he oal ferly rae s gong o sablze n fuure? And secondly, wha s he funcon ha mus o use o predc he oal ferly rae? These wo quesons were answered usng he Sable Bounded Theory, whch allowed us o prove he exsence of a sablzer value and o calculae. Also, we found he funcon ha allowed us o do he predcons of he oal ferly rae. Fnally, we acheved he surprsng resul ha he sablzng funcon was no a logsc funcon, as n many sudes. I can be verfed by havng daa for when he rae begns o lower slowly, hen when lowers rapdly, and fnally when reurns o lowerng slowly. Unforunaely, n hs exercse we only had daa of when he rae drops rapdly and when falls slowly, whch can be a lmaon. Conclusons In Mexco, for he perod , he behavor of he mean of he oal ferly rae hrough me s governed by a mahemacal funcon ha depends on me. The funcon s deermned by hree parameers, he sablzer value K=1.98 and he quckness parameers and m= These parameers can change accordng o he counry or analyzed perod. Ths model canno be used o know he behavor of oal ferly rae n he pas, because accordng o hs model when we go back n me, he oal ferly rae grows and grows, whch s no possble. In Mexco for explanng he evoluon of he oal ferly rae hrough me, he demographers have used a logsc funcon. However, hs paper proves ha hs s an error, snce wo sablzer values do no exs as he logsc funcon needs. Alhough hs exercse was done wh daa from Mexco, s mporan o make clear ha he Sable Bounded Theory can be appled o any counry where daa on he oal ferly rae are avalable. I s necessary o warn ha he resuls of hs paper are based on he assumpon ha he socal, economc and polcal condons wll connue whou change. If hs assumpon s no fulflled, he forecass wll no be rue. Also, s necessary o warn ha he mahemacal modelng of realy s based on many assumpons, and he heorecal resuls are rue only f he assumpons are fulflled. Thus, s necessary o expend grea effor o prove ha he assumpons are rue. Fnally, any exercse o predc he fuure s exposed o a lo of error sources: wrong daa, false assumpons and hypoheses, ncorrec models, and so on. Therefore, s necessary o denfy all possble error sources, and hen ulze mehodologes ha mnmze hose errors. The Sable Bounded Theory s an example of ha. 14

15 Ahens Journal of Socal Scences XY References Alkema L, Rafery AE, Gerland P, Clark SJ, Pelleer F, Buener F, Helg GK (2011) Probablsc projecons of he oal ferly rae for all counres. Demography 48(3): Rereved from hps://b.ly/2hj1p33. [Accessed 9 Aprl 2017]. do: /s Alkema L, Gerland P, Rafery A, Wlmoh J (2015) The Uned Naons Probablsc Projecons: An nroducon o demographc forecasng wh unceranly. Foresgh (Colcheser, V), 2015(37): Rereved from hps://b.ly/2qtsohu. [Accessed 10 Aprl 2017]. Ashraf QH, Wel DN, Wlde J (2013) The Effec of ferly reducon on economc growh. In Populaon and Developmen Revew 39(1): do: /j x Coordnacón del Programa Naconal de Planfcacón Famlar (NSPUCM) (1979) Encuesa Naconal de Prevalenca en el Uso de Méodos Anconcepvos. Resulados naconales [Survey of Prevalence n he Use of Conracepve Mehods. Naonal resuls]. Méxco, p Dreccón General de Planfcacón Famlar (1989) Encuesa Naconal sobre Fecunddad y Salud 1987 [Naonal Survey on Ferly and Healh 1987]. Mexco Mnsry of Healh. Undersecreara of Healh Servces, p Draper J, Smh, W (1966) Appled Regresson Analyss. New York: John Wley & Sons. González-Rosas J (2012) La Teoría Esable Acoada: Fundamenos, concepos y méodos, para proyecar los fenómenos que no pueden crecer o decrecer ndefndamene [The Sable Bounded Theory: Fundamenals, conceps and mehods, o projec phenomena ha canno grow or decrease ndefnely]. Saarbrucken, Germany: Spansh Academc edoral. INEGI (2017). Prncpales resulados de la Encuesa Inercensal de 2015 [Man resuls of he 2015 Inercensal Survey]. Rereved from hps://b.ly/2qpjpyw. [Accessed 18Aprl 2017]. Lehold L (1973) El Cálculo: Con geomería analíca [The calculaon wh analyc geomery], 2 nd ed. Mexco: Harla S.A. Of C.V. Luz W (2006) Ferly raes and fuure populaons rends: wll Europe s brh rae recover or connue declne? Inernaonal Journal of Andrology. Rereved from hps://b.ly/2hoxar. [Accessed 11 Aprl 2017]. do: /j x. Medh J (1981) Sochasc Processes, 2 nd ed. New York: John Wley & Sons. Parda-Bush V (2008) Proyeccones de la poblacón de Méxco, de las endades federavas, de los muncpos y de las localdades [Projecons of he populaon of Mexco, he saes, muncpales and locales ]. Mexco: Mehodologcal documen. Naonal Populaon Councl. Secrearía de Programacón y Presupueso (1979) Encuesa Mexcana de Fecunddad. Prmer nforme naconal [Mexcan Ferly Survey. Frs naonal repor]. Mexco, p Soboka T, Zeman K, Basen S (2015) The Low Ferly Fuure? Projecons Based on Dfferen Mehods Sugges Long-erm Perssence of Low Ferly. Rereved from hps://b.ly/2hjlddh. Uned Naons, Deparmen of Economc and Socal Affars, Populaon Dvson (2015). World Ferly Paerns Daa Bookle (ST/ESA/ SER.A/370). Rereved from hps://b.ly/1svrzyo. [Accessed 10 Aprl 2017]. Wlye CR (1979) Dfferenal equaons. Mexco: McGraw Hll. 15

16 Vol. X, No. Y González-Rosas e al.: The Sable Bounded Theory... Annex Table 1. Toal Ferly Rae Projecons n Mexco, Year Tme Toal Ferly Rae Year Tme Toal Ferly Rae Source: Own calculaons based on equaon 12 16