Modelling and projecting the postponement of childbearing in low-fertility countries

Transcription

1 of childbearing in low-fertility countries Nan Li and Patrick Gerland, United Nations * Abstract In most developed countries, total fertility reached below-replacement level and stopped changing notably. Contrary to the almost stationary fertility level, the mean age of childbearing () is increasing markedly, reflecting a more general postponement transition that may correspond to fundamental shifts for society (Lee and Goldstein, 003, Billari, 005). The postponement of childbearing, however, includes changes in fertility at all reproductive ages, which are more complicated than the increase of. Using the idea of the Lee-Carter (99) method that deals with mortality, we show that the changes in the age pattern of fertility can be well described by a single variable, which leads to simple ways of projection. We discuss some application issues using the data from Italy, and describe a condition under which the model is epected to work well for other countries. The model and its estimations Most fertility studies deal with age-specific fertility rate (ASFR), which sums to total fertility (TF) over the reproductive ages. To model the age pattern of fertility, we focus on the proportionate ASFR, which is ASFR/TF, and sums to over the reproductive ages. Denote the proportionate ASFR at age and time t by Fp(,, and denote the overtime average of Fp(, by a(, the model can be written as Fp (, a( + b( k(. () where b( is the rate of change by age groups and k( is the overall time trend. One may see immediately that the right-hand side is identical to the Lee-Carter (99) model. In fact, what we really borrowed from the Lee-Carter method is the idea, which is * Population Estimates and Projection Section, United Nations Population Division, Dept. of Economic and Social Affairs, United Nations, New York, NY 007, USA. The views epressed in this paper are those of the authors and do not necessarily reflect those of the United Nations. Its contents has not been formally edited and cleared by the United Nations.

2 to convert the task of dealing with a group of variables (Fp(, at different ages) to that of handling a single variable k(. Parameters b( and variable k( in model () can be estimated in various ways, and we discuss two below. The first estimation is obtained using the ordinary least squares (OLS). To see this, let b( =. () Thus, k( is the difference between the observed at time t and the given by a(: Fp(, a( = k( b( = k(. (3) Since k( is known, b( is solvable using OLS to minimise t [ Fp(, a( b( k( ] with constrain (): Fp(, k( t b( = k (. (4) t An advantage of the first estimation is that the model is identical to the observed value at any time. The disadvantage of first estimation is, however, that the difference between the observed and model Fp(, has no constraint on valid bounds. The second estimation is obtained using the singular value decomposition (SVD), which provides the values of b( and k( that minimises t [ Fp(, a( b( k( ] (see, Lee and Carter, 99). For their convenience, Lee and Carter scaled the b( to sum to over all. For our convenience, we scale the b( as in (), so that k( is the difference between the model at time t and the given by a(. The advantage of the second estimation is that the sum of the squared difference between the observed and model Fp(, is minimised. A disadvantage of the second estimation, however, is that there are differences between the observed and model values of, which is zero using the first estimation.

3 Some application issues Denote by Fpm(, the model value of Fp(,. The eplanation ratio, which indicates the proportion of the variance of Fp(, eplained by Fpm(,, is defined as [ Fp(, Fpm(, ] t R = [ Fp(, a( ]. (5) t Using the data on Fp(, in Table, we have R=0.85 (SVD) and R=0.75 (OLS). Table. Italian proportionate ASFR Sources: UNSD and Eurostat Thus, as we epect, SVD worked better than OLS did in terms of describing Fp(,. On the other hand, as is shown in the third panel of Figure, SVD cannot perfectly fit the (, which is described eactly by OLS. Therefore, which estimation to use depends on which variable, Fp(, or Mac(, is more important to the user. 3

4 Why does the model work well? Putting in the continuous version, () yields [ Fp(, a( ] { [ Fp(, a( ] d dt [ Fp(, a( ] k( d dt d dt [ Fp(, a( ]} 0. k(, Thus, the condition for () to work well is that the rates of change in [Fp(,-a(] are similar at different ages. In fact, the changes of age pattern of fertility that we concern are two rotations. The first rotation is caused by the reduction of childbearing at older ages, in which [Fp(,-a(] rises at younger and drops at older ages. And the second rotation is due to postponing childbearing, in which [Fp(,-a(] drops at younger and rises at older ages. When such rotations take place at the rates that are similar over ages, (6) holds. Therefore, we may epect () to work well not only for the data of Italy, but also those when the rotations take place evenly over age. Turning to projection, the above model will yield a maimum, namely m, older than which the model will produce negative values for Fp(, at some (6) 4

5 younger ages. This is because that b( is negative at younger ages, which will make Fp(, negative when k( rises to a certain level Km: a( Km = min[ ], b( < 0. (7) b( Since k( differs with ( by a constant According to () and (), the maimum is written as m = a( + Km. (8) a (, Km leads to the m. Using data in , the m is estimated as 3.0 by SVD, or 3. by OLS. Given the model (SVD) values of ( and m as eample, there are various simple ways to project future (. Among these, the one shown in the third panel is perhaps the simplest, in which the ( converges to m eponentially at the pace measured from its last two values. When ( is projected, so are the Fp(, using (), as can be seen in the last panel of Figure. When the m given by (8) looks too small to be plausible, for eample of many East European countries, we suggest use a plausible maimum that may be taken from other countries, and replace the subsequent negative values of the projected Fp(, at a certain by its last positive value. By doing so, we projected a more plausible trajectory for the, and stopped following the modelled projection of Fp(, when it becomes negative. 5

6 References Lee, R. D. and L. Carter. 99. Modelling and Forecasting the Time Series of U.S. Mortality. Journal of the American Statistical Association 87: Lee, R. and J.R. Goldstein Rescaling the life cycle: Longevity and proportionality. Population and Development Review, 9: Billari, F Partnership, childbearing and parenting: Trends of the 990s. In The new demographic regime: Population challenges and policy responses. United Nations: New York and Geneva. 6

7 0/7/009 Modelling and projecting the postponement of childbearing in low-fertility countries Nan Li & Patrick Gerland IUSSP 009 Marrakech, Morocco ( Oct. 009) Session 64: Timing of fertility and family transitions in Europe United Nations Population Division Estimates and Projection Section Outline Background Data Estimation and forecasting model Results and findings The views and opinions epressed in this slides are those of the author and do not necessarily represent those of the United Nations. The contents has not been formally edited and cleared by the United Nations. The designations employed and the material in these slides do not imply the epression of any opinion whatsoever on the part of the Secretariat of the United Nations concerning the legal status of any country, territory or area or of its authorities, or concerning the delimitation of its frontiers or boundaries IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org)

8 0/7/009. Issue: Total fertility vs. Mean Age of Childbearing ( ) keeps rising while TF stagnates below replacement level or fluctuates toward replacement level Italy: Covariance [TF,]= Western Europe Italy ( ): Covariance[TF,] = TF IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org)

9 Austria ( ): Covariance[TF,] = Finland ( ): Covariance[TF,] = Czech Republic ( ): Covariance[TF,] = Bulgaria ( ): Covariance[TF,] = Denmark ( ): Covariance[TF,] = Iceland ( ): Covariance[TF,] = Hungary ( ): Covariance[TF,] = France ( ): Covariance[TF,] = Estonia ( ): Covariance[TF,] =- Ireland ( ): Covariance[TF,] = Poland ( ): Covariance[TF,] = Luembourg ( ): Covariance[TF,] = Norway ( ): Covariance[TF,] = Lithuania ( ): Covariance[TF,] = Netherlands ( ): Covariance[TF,] = Sweden ( ): Covariance[TF,] = Slovakia ( ): Covariance[TF,] = Switzerland ( ): Covariance[TF,] = United Kingdom ( ): Covariance[TF,] = Slovenia (98-007): Covariance[TF,] = Romania ( ): Covariance[TF,] = /7/009 Southern Europe Greece: Covariance [TF,]=-0. Spain: Covariance [TF,]=-0. Greece ( ): Covariance[TF,] =-0. Spain ( ): Covariance[TF,] =-0. Portugal: Covariance [TF,]=-0.4 Portugal ( ): Covariance[TF,] =-0.4 Austria Cov=-0.07 Western Europe Denmark France Luembourg Netherlands Switzerland Cov=0.0 Cov= Cov=0.5 Cov=0.07 Cov=- Northern Europe Finland Iceland Ireland Norway Sweden U.K. Cov=- Cov=0.03 Cov=- Cov= Cov=-0.03 East-Central Europe Czech Rep. Hungary Poland Slovakia Slovenia Cov=-0.46 Cov=-0.9 Cov=-0.38 Cov=-0.5 Cov=-0.0 Cov=-0. Eastern Europe Bulgaria Estonia Lithuania Romania Cov=-0.8 Cov=- Cov=- Cov=- IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 3

10 0/7/009. Data Data sources Eurostat database ( 5 countries with annual fertility rates by single age for the last 5 years or more. Western Europe: Austria, Denmark, France métropolitaine, Luembourg (Grand-Duché), Netherlands, Switzerland. Northern Europe: Finland, Iceland, Ireland, Norway, Sweden, United Kingdom 3. Southern Europe: Greece, Italy, Portugal, Spain 4. East-Central Europe: Czech Republic, Hungary, Poland, Slovakia, Slovenia 5. Eastern Europe: Bulgaria, Estonia, Lithuania, Romania IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 4

11 0/7/ Model and estimation strategy Age pattern of fertility modelling F(, = age-specific fertility rate at age and time t TF ( t ) 50 5 F (, t ) Fp(, = proportionate age-specific fertility rate at age and time t F (, Fp (, TF ( with 50 5 Fp (, Italy: Fp(, (=7.6) 005 (=.8) IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 5

12 0/7/009 Age pattern of fertility modelling Fp (, a( b( k ( Identical to Lee, R. D. and L. Carter. 99. Modelling and Forecasting the Time Series of U.S. Mortality. Journal of the American Statistical Association 87: a( = over-time average of Fp(, b( = rate of change by age groups k( = overall time trend a( = average (pas age pattern IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 6

13 0/7/009 Two ways to estimate b( and k( () Ordinary Least Squares (OLS) rescaling b( to get b ( k( = difference between observed at time t and given by a( Fp (, a ( k ( b ( k ( and b( t Fp (, t ) k ( t ) t k ( t ) Solved by minimizing t [ Fp (, a( under constraint that b( k ( ] b ( Two ways to estimate b( and k( () Singular Value Decomposition (SVD) by minimizing: [ Fp (, a( b( k ( ] t rescaling b( to get b ( k( = difference between the model at time t and the given by a( IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 7

14 ( Fp(, b( k( Modelling and projecting the postponement 0/7/009 A.Italy ( ): Rate of change by age b( B.Italy ( ): Time Trend k( 0.0 b(.5 k( OLS SVD -0.0 OLS -.5 SVD Age Year t C.Italy ( ): Mean Age of Childbearing ( D.Italy: ASFR density distribution PASFR( (005) = Fp(, (OLS proj.) 050 (SVD proj.) 3 Maimum : 0.07 m: OLS=3.5 SVD= OLS SVD Observed OLS Proj. mac050=3.4 SVD Proj. mac050= Year t Age OLS vs. SVD: Pros and cons () OLS: best to fit, but estimates of b( and k( done sequentially, and difference between the observed and model Fp(, has no constraint on valid bounds () SVD: best to fit overall age pattern, estimates of b( and k( done simultaneously with constraints on valid bounds but differences between observed and model values of IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 8

15 0/7/009 Goodness of fit Eplanation Ratio (R) = proportion of the variance of Fp(, eplained by Fpm(, which is the model value of Fp(, [ Fp (, t ) Fpm (, t )] R t [ Fp (, t ) a ( ] t Maimum m = age limit at which the model will produce negative values for Fp(, at younger ages because b( < 0 at younger ages and k( rises to a certain level Km: Km min[ a( ], b( b( Since k( differs with ( by a constant Km leads to 0 a ( m a ( Km IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 9

16 0/7/009 K( projection from to 050 Only k( term in model is time dependent and needs to be projected Since k( = difference between the model at time t and the given by a( Project ( to converge toward m eponentially using trend for past 0 years. 4. Results and Findings IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 0

17 0/7/009 Model fitting performance () very good fit for most countries (>80% or even 90% variance eplained) () SVD fit outperforms OLS fit Country minyear mayear CovTfMac R(OLS) R(SVD) Western Europe Austria Denmark France Luembourg Netherlands Switzerland Northern Europe Finland Iceland Ireland Norway Sweden United Kingdom Southern Europe Greece Italy Portugal Spain East-Central Europe Czech Republic Hungary Poland Slovenia Slovakia Eastern Europe Bulgaria Estonia Lithuania Romania a( = average age pattern of fertility 0.09 A. Western Europe Austria (=7.6) 0.09 B. Northern Europe Finland (=9) 0.09 C. Southern Europe 0.07 Switzerland (=9.3) Denmark (=.8) France (=.7) Luembourg (=.7) Netherlands (=9.5) 0.07 Ireland (=.4) Iceland (=.) Norway (=.5) Sweden (=9.) United Kingdom (=) 0.07 Spain (=9.6) Greece (=7.9) Italy (=9.4) Portugal (=) D. East-Central Europe E. Eastern Europe Czech Republic (=6) Hungary (=6.4) Poland (=7.3) Slovenia (=7.) Slovakia (=6) Bulgaria (=4.5) Estonia (=6.6) Lithuania (=6.6) Romania (=5.6) IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org)

18 0/7/009 b( = rate of change by age A. Western Europe B. Northern Europe C. Southern Europe Austria Switzerland - Denmark France Luembourg Netherlands Finland Ireland - Iceland Norway Sweden - United Kingdom Spain Greece - Italy Portugal - D. East-Central Europe E. Eastern Europe Czech Republic Hungary Poland Slovenia Bulgaria Estonia Lithuania Romania - Slovakia - k( = overall time trend A. Western Europe B. Northern Europe C. Southern Europe Austria Switzerland Denmark France Luembourg Netherlands Finland Ireland Iceland Norway Sweden United Kingdom Spain Greece Italy Portugal D. East-Central Europe E. Eastern Europe Czech Republic Hungary Poland Slovenia Slovakia Bulgaria Estonia Lithuania Romania IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org)

19 0/7/009 Mean Age of Childbearing ( A. Western Europe B. Northern Europe C. Southern Europe 9 Austria Switzerland 7 Denmark 6 France Luembourg 5 Netherlands Finland 7 Ireland Iceland 6 Norway 5 Sweden 4 United Kingdom Spain 7 Greece 6 Italy 5 Portugal D. East-Central Europe 33 E. Eastern Europe Czech Republic 7 Hungary 6 Poland 5 Slovenia 4 Slovakia Bulgaria Estonia Lithuania Romania Mean Age of Childbearing Maimum Mean Age Decennial Modal Age Median Age Country Increase Western Europe Austria Denmark France Luembourg Netherlands Switzerland Northern Europe Finland Iceland Ireland Norway Sweden United Kingdom Southern Europe Greece Italy Portugal Spain East-Central Europe Czech Republic Hungary Poland Slovenia Slovakia Eastern Europe Bulgaria Estonia Lithuania Romania IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 3

20 p(90)-p(0 p(90)-p(0 p(90)-p(0 p(90)-p(0 p(90)-p(0 0/7/ Dispersion of the fertility age distributions Italy 4 Mean Mode p(0) 38 0 Median p(90) p(90) = Age for upper 90 percentile p(0) = Age for lower 0 percentile Mean Mode p(0) Median p(90) Czech Republic Median age vs. p(90)-p(0) age dispersion A. Western Europe Austria Switzerland Denmark France Luembourg Netherlands Median D. East-Central Europe Finland Ireland Iceland Norway Sweden United Kingdom B. Northern Europe Median Czech Republic Hungary Poland Slovenia Slovakia Median Spain Greece Italy Portugal C. Southern Europe Median E. Eastern Europe Bulgaria Estonia Lithuania Romania Median IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 4

21 Fp(, 0/7/009 Projected age pattern in 050 A. Western Europe B. Northern Europe C. Southern Europe 0.0 Austria Switzerland Denmark France Luembourg Netherlands 0.0 Finland Ireland Iceland Norway Sweden United Kingdom 0.0 Spain Greece Italy Portugal D. East-Central Europe Czech Republic Hungary Poland Slovenia Slovakia 0.0 E. Eastern Europe Emerging of bi-modal age distributions Bulgaria Estonia Lithuania Romania Emerging of bi-modal age distributions United Kingdom (OLS proj.) 050 (SVD proj.) Age IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 5

22 ( Fp(, b( k( Modelling and projecting the postponement 0/7/009 b( = rate of change by age B. Northern Europe B. Northern Europe Ireland United Kingdom Lack of decline (or even increase) in fertility under age 0 in UK, Ireland, Slovakia, Bulgaria, Romania, etc Finland Ireland Iceland Norway Sweden United Kingdom Challenges with East-Central Europe and Eastern Europe A.Czech Republic ( ): Rate of change by age b( B.Czech Republic ( ): Time Trend k( 3.5 OLS SVD OLS 0 - b( < 0 SVD Age Year t C.Czech Republic ( ): Mean Age of Childbearing ( D.Czech Republic: ASFR density distribution PASFR( (007) =.8 Maimum : m: OLS=9.6 SVD=9.5 artificial stagnation OLS SVD Observed (OLS proj.) 050 (SVD proj.) Fp(, < 0 due to b( < 0 OLS Proj. mac050= SVD Proj. mac050= Year t Age IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 6

23 b( ( ( Fp(, k( Fp(, 0/7/009 Additional constraints If m <= latest observed or implausibly low, compute m using eperience from other countries m = latest * ratio from all countries (ecept Central & Eastern Europe) of [average(m) / average(latest )] = latest *.05 Replace subsequent negative values of the projected Fp(, at a certain by its last positive value Eample: Czech Republic Age A.Czech Republic ( ): Rate of change by age b( C.Czech Republic ( ): Mean Age of Childbearing ( (007) =.8 Maimum : m: OLS=9.6 SVD=9.5. Unconstraint OLS OLS SVD SVD - Observed OLS Proj. mac050=9.6 Age SVD Proj. mac050= Year t OLS Year t B.Czech Republic ( ): Time Trend k( SVD D.Czech Republic: ASFR density distribution PASFR( (OLS proj.) 050 (SVD proj.) Year t Age 3 C.Czech Republic ( ): Mean Age of Childbearing ( 0. D.Czech Republic: ASFR density distribution PASFR( 9 (007) =.8 Maimum : (input=.6) m: OLS=9.6 SVD= (OLS proj.) 050 (SVD proj.) 7. Constraint Constraint: OLS = min(pasfr)>=0 SVD = min(pasfr)>=0 6 5 OLS SVD Observed OLS Proj. mac050=.6 SVD Proj. mac050= Year t Age IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 7

24 0/7/009 Mean age in 050 Sensitivity analysis Country n=0 n=5 n=5 Western Europe Austria Denmark How much does the length of the reference period matters to project k(? -> use the latest 0, 5,, 5 years? France Luembourg Netherlands Switzerland Northern Europe Finland Iceland Ireland Norway Sweden United Kingdom Southern Europe Greece Italy Portugal...0 Spain East-Central Europe Czech Republic Hungary.3.. Poland Slovenia Slovakia Eastern Europe Bulgaria Estonia.3.3 Lithuania Romania Netherlands: trend in Austria Switzerland Denmark France Luembourg Netherlands k( Western Europe ( Western Europe Austria Switzerland Denmark France Luembourg Netherlands IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 8

25 0/7/ Netherlands: trend in k( k( Netherlands k( observed k( projected with n=0 k( projected with n= k( projected with n= Sensitivity of the length of period of reference used to project trend in k( n= 0, 5,, 5 on age pattern of fertility in 050 Limited effect, overall very robust Age-Specific Fertility Pattern 007 n=0,050 n=5,050 n=5, Conclusion Overall approach robust and fleible to project fertility age patterns for below-replacement countries SVD fitting performs better than OLS to find b( and k( High consistency within and between regions for future age patterns of fertility Identification of special cases:. Adolescent fertility not declining in some countries potentially leading to future bi-modal age distributions ;. Special case of Central and Eastern Europe postponing childbearing through very pronounced declines at younger and rises at older ages requiring additional projection constraints. IUSSP Session 64 - Nan Li (li3@un.org) & Patrick Gerland (gerland@un.org) 9