Effects of Immigration on Measuring Cohort Fertility

Transcription

1 Effects of Immigratio o Measurig Cohort Fertility Felix Roessger Max Plack Istitute for Demographic Research, Rostock (Germay) roessger@demogr.mpg.de Exteded abstract for submissio to EPC 212 (Coferece Topic: Data ad Methods) October 14, 211 Abstract I this paper we examie the effects of immigratio o the cohort total fertility rate (CTFR) as covetioal measure for completed cohort fertility. As almost all covetioal demographic measures it was desiged for closed populatios ad ca be cosequetly biased, if it is applied to a ope populatio. We show that this bias arises from the fact that the immigrat populatio is ot icluded i the populatio uder exposure properly. Cosequetly, we propose a measure adapted for immigratio, which treats migrats as if they always had bee i the cosidered populatio. However, this ew measure is restricted due to its high requiremets o detailed fertility data. For this reaso, a further iterative procedure is examied, which models the relevat process uder certai assumptios but with less data requiremets.

2 Itroductio Covetioal demographic measures, such as life expectacy at birth or the total fertility rate, are usually desiged for closed populatios. Presto ad Coale (1982) showed already 3 years ago formally that these measures ca be adapted to ope populatios by icludig age-specific migratio rates as well. However, applicatios or advacemet of such measures are hard to fid i the demographic literature. Relevat research questios from a methodological poit of view i this cotext ca be thought of i two directios: (1) How ca covetioal measures be misleadig if they are applied to ope populatios? (2) How ca covetioal measures be misleadig if they are applied to migrat populatios? The first questio was addressed, for istace, with regard to the et reproductio rate. Through providig results for a (et) et reproductio rate, that is accoutig for outmigratio, Espeshade (1982) showed that the covetioal measure could suggest o-realistic estimates of replacemet-level fertility if outmigratio is excluded. Presto ad Wag (27) advaced this idea by suggestig a et reproductio rate that allows for et migratio. The secod questio was addressed by Toulemo (24) ad Parrado (211), who exhibited the issue that fertility of immigrats is more closely liked to the timig of migratio tha to the age of the migrats. Cosequetly, just summig up age-specific fertility rates could yield misleadig estimates for the total fertility rate (TFR) of differet immigrat groups, sice ot the etire fertility history of them is icluded. Therefore, the authors suggested adapted measures that iclude the etire fertility history of immigrats, which lead to dramatically differet estimates for the TFRs of immigrats i the case of Frace (Toulemo 24) or for Hispaics i the Uited States (Parrado 211). This study deals more with the first research directio but is also related to the fidigs of Toulemo (24) ad Parrado (211). We will ivestigate how the ot icluded part of the fertility history of immigrats matters for a covetioal measure of cohort fertility the cohort total fertility rate (CTFR). Usually we thik of the TFR as distorted ad CTFR as udistorted. This study reveals that also the CTFR ca be biased, amely whe it is applied o ope populatios. Towards a immigratio adapted measure of cohort fertility Let us call f(x) the age-specific fertility rates of the total populatio at age x ad N(x) the total age-specific populatio itself. A superscript or i labels the respective umbers for atives or immigrats ad is the age at which the reproductive period eds. The covetioal CTFR ca the be writte as f (x) N(x) (1) CTFR = dx. N(x) 1

3 If we cosider ow immigratio with I(x) beig the immigrats added to the cohort at age x with the survival fuctio l(x), we ca express how the covetioal CTFR deals implicitly with immigrats ad their fertility as follows 1 : i l(x) f (x) N (x) + f (x) I(a) da l(a) (2) CTFR = dx. x l(x) N (x) + I(a) da l(a) We see that immigrat fertility is oly cosidered for immigrats who are preset i the aalyzed populatio. At ages where the immigrats are ot preset, the ative fertility schedule gets the full weight. However, if the fertility schedule of immigrats is differet from the oe of the atives, this could bias our measure. If for example immigrats have exactly the same fertility schedule as atives, but shifted by oe year to older ages, we would obtai a CTFR-value that is higher tha the CTFRs of atives ad immigrats. The reaso is that the used correspodece betwee evets ad exposure is broke, because immigrats, i the case of delayed fertility, do ot cotribute exposure before they have arrived. This becomes particularly evidet whe we look at a small sythetic example. Let us assume a populatio with oly two ages where everybody gets a child. The atives have their child at the first age ad the immigrats at age two. If the immigrats add to the cohort oly at age two, we observe o the total populatio level a age specific fertility rate of oe child per wome at the first age ad a age-specific fertility rate that is smaller tha oe (but bigger tha zero) at the secod age. This leads to a covetioal CTFR that idicates that the whole populatio has more tha oe child per wome. Table 1 illustrates this for a populatio of te atives ad te immigrats for which we would estimate a CTFR of 1.5 childre per wome although both ivolved populatio subgroups have oly oe child durig their reproductive period. Table 1: Cohort fertility statistics for a model populatio with two age groups Births to Births to Total Age Natives Immigrats Total Populatio Natives Immigrats Births f (x) f i (x) f(x) CTFR For a CTFR-measure that icludes both parts of the fertility history of immigrats the oe before ad the oe after migratio, we suggest treatig immigrats withi this calculatio as if they had bee always i the populatio but with their specific fertility rates: x 1 The latter a also deotes age, but is labeled differetly to x to avoid cofusio i the double itegral. 2

4 f (x) N (3) CTFR* = dx. N i (x) + f (x) (x) + I(a)da l(x) I(a)da l(x) Withi the sythetic example i Table 1 this corrected measure, called CTFR*, would yield values of.5 for the age-specific fertility rates at age oe ad age two. Cosequetly, the value of the CTFR would be estimated with 1. childre per wome, which is exactly the figure from both populatio subgroups. Ufortuately, the detailed data required for Formula (3) is rarely available. Age-specific fertility rates are usually ot provided subdivided for atives ad foreig bor. Hece, it appears useful to cosider models of this process which ca be applied uder certai assumptios with less data requiremets. Oe optio will be demostrated i the followig example: If immigratio starts oly at the secod reproductive age, the observed age-specific fertility rate at the first reproductive age is equal to the oe of the atives. If we the have reaso to assume that the fertility schedules of atives ad immigrats are similar, but postpoed by a certai time-lag (δ) for the immigrats due to the evet of the move, we could get the ative fertility-schedule back with help of a iterative procedure 2 - with f () as iitial value for estimatig f (1), f (1) as iitial value for estimatig f (2) ad so o. For this procedure we oly eed the populatios of foreig bor ad atives for estimatig the agespecific shares of immigrats w(x), but o subdivided fertility statistics. Oly the observed age-specific fertility rates for the total populatio are still required: (4) f f (x) f (x δ) w(x) (x) =. 1 w(x) If the fertility schedules of atives ad immigrats are idetical but oly shifted, the the CTFR for atives equals our corrected measure CTFR*: (5) CTFR = CTFR* = f (x) dx.. We ca test this with our example from Table 1 ad reduce our kow iformatio by the subdivided fertility statistics for atives ad foreig bor: Table 2: Cohort fertility statistics for a model populatio with two age groups (reduced iformatio) 3 Total Age Immigrats Total Populatio Births w(x) f(x) CTFR The detailed derivatio is available i full paper. 3 I additio to Table 1 we calculated the shares of immigrats here which is simple I(x)/N(x) i our example. 3

5 With help of Formula (4) we ca estimate the age-specific fertility rate of atives at age 2 which yields the true value of zero ad we kow that the age-specific fertility rate of atives at age 1 is equal to the observed oe at this age (i.e. 1.). Our estimate for the CTFR of immigrats ad cosequetly for the CTFR* would tha be the correct value of 1. childre per wome. Coclusio We showed that the covetioal measure of completed cohort fertility, the CTFR, ca be misleadig uder the presece of immigratio, but also that it is possible to correct it for this bias. The magitude of this distortio depeds, o the oe had, o the share of immigrats i a cosidered populatio ad, o the other had, o the differeces i the fertility schedules of atives ad immigrats. Cosequetly, we expect relevat differeces especially for smaller coutries or regios where immigratio plays a importat role such as Switzerlad or Israel. I the case of Israel fertility data by place of birth is available - at least for the Jewish populatio (see Cetral Bureau of Statistics 1996). Thus, the Israeli cotext appears a good oe to apply the suggested correctio with real empirical data. But sice the availability of data for this research directio is usually very limited, the search for models with less data-requiremets ad their testig appears as essetial part withi this research project ad withi further research o this topic i geeral. Refereces Cetral Bureau of Statistics Statistical Abstract - Fertility Rates by Age, Religio ad Cotiet of Birth. [ Espeshade, T.J "A (et) et reproductio rate?" Iterco, Populatio Referece Bureau 1(11/12):8-1. Parrado, E "How High is Hispaic/Mexica Fertility i the Uited States? Immigratio ad Tempo Cosideratios." Demography 48(3): Presto, S.H.ad A.J. Coale "Age Structure, Growth, Attritio, ad Accessio: A New Sythesis." Populatio Idex 48(2): Presto, S.H.ad H. Wag. 27. "Itrisic Growth Rates ad Net Reproductio Rates i the Presece of Migratio." Populatio ad Developmet Review 33(4): Toulemo, L. 24. "Fertility amog immigrat wome: ew data, a ew approach." Populatio & Societies(No. 4). 4