In order to describe age-specific fertility curves, different theoretical approaches are employed. By modelling a time-variant fertility curve, it is possible to forecast reproductive behaviour of Estonian women in the future. The approach provided here is a novel way to forecast fertility

Vähi, Mare

Journals from University of Tartu

FERTILITY MODELLINGMare VähiInstitute of Mathematics and Statistics, University of Tartu, Tartu, EstoniaABSTRACTIn order to describe age-specific fertility curves, different theoretical approaches are employed. By modelling a time-variant fertility curve, it is possible to fore-cast reproductive behaviour of Estonian women in the future. The approach provided here is a novel way to forecast fertility.Keywords: age-specific fertility rate; beta distribution; forecast.For years, modelling of fertility curves by theoretical distribution has been used to describe reproductive patterns of women in child-bearing age. Several math-ematical models fi t adequately with fertility curves. Th e curve-fi tting approach is based on the age-specifi c fertility rates.Th e age-specifi c fertility rates are calculated on the basis of the following data: the number of women in child-bearing age (aged 15–49) and the number of children born by these women. Using fx to designate the number of women aged x years (x = 15, 16, … 49) and bx to designate the number of children born within a year by women aged x, the fertility rate for the women aged x may be calculated using the following equation:vx=bx / fx .To provide a more explicit interpretation, the fertility rate is oft en expressed as the calculated value multiplied by 1,000, and the result represents then the number of children born within a year for 1,000 women.Th e new approach to stochastic forecast of the age-specifi c fertility rates can be summarized as follows. Fertility modellingMare Vähihttps://doi.org/10.12697/poa.2017.26.1.09Papers on Anthropology XXVI/1, 2017, pp. 107–114108  |  Mare Vähi(1) Fit the suitable family of distributions for modelling the fertility rates. (2) Estimate the parameters of distribution. (3) Forecast the parameters using the time series model. (4) Use the forecast parameters to forecast the model for age-specifi c fertility rates by one-year age groups.Th e methodology was applied to Estonian fertility data. Empirical data were taken from the Statistical Database of Statistics Estonia.Th e model was designed based on the data for years between 1991 and 2009. Figure 1 shows the variation of respective fertility rates.02040608010012014016015 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49Age, yearsRate1991-1992 1993-1996 1997-2000 2001-2009Figure 1. Age-specifi c fertility rateIn the above-mentioned range of years, four periods may be distinguished: years 1991–1992, 1993–1996, 1997–2000 and 2001–2009. In the fi rst period, fertility rates are high in all age groups and the highest in the group of women aged 20 to 22. In the second period (1993–1996), the delay in giving births may be observed. Th e number of young mothers (aged 20–24) falls sharply, but the number of mothers in the age group 25–36 also decreases. In the period of 1997–2000, the number of births still decreases in the age group 20–24, but the fall is not as sharp as in the earlier period. In the older groups (starting from the age of 25 years), the number of births remains on the level of 1993. An additional trend, however, can be noticed – the number of births increases in the group of women aged 31 and 32.    Fertility modelling  |  109Compared with the year 1996, the number of mothers has decreased even more by the year 2000 in younger age groups, whereas it has increased in older age groups, which suggests that the women who delayed child births in the earlier years started to give birth. Consequently, the number of mothers has grown in the age group 27–38.In the last period or in this century, the number of mothers has further declined in the age group 18–24, but the increase in the number of births has been considerable in the age group 25–42. In addition, the curves that were inclined to the left  in the fi rst periods are becoming more and more symmetrical. A symmetrical, bell-shaped fertility curve is characteristic of the fertility curves of several European countries.As shown in Figure 1, the fertility curves of the studied periods diff er considerably. Th e next step is to fi nd theoretical distribution so as to describe the fertility curves. Keeping in mind the aim of using a distribution appropriate for describing fertility curves to forecast the fertility, it would be benefi cial to describe all the curves within the observed period by means of the same type of distribution. Th e researchers studying the fertility curves of diff erent countries have used various distributions for approximation. Th e most widely used distri-butions are the gamma distribution, the beta distribution and the Hadwiger distribution. Th e latter has been employed, for example, in Sweden, Denmark, France and Austria. Th e beta distribution has been used in various European countries, whereas a mixture of distributions has been used in Ireland and Great Britain. By using the said distributions, we also tried to approximate the Estonian data. Th e gamma density function is expressed as follows:  , wherein β and γ represent the distribution parameters and Γ(γ) is the gamma function. Hadwiger function is expressed as  , 110  |  Mare Vähiwherein α, β and γ represent estimated parameters and x is the age of the mother at childbirth.Th e following interpretation of the parameters has been suggested: the param-eter α is associated with the general fertility rate, the parameter β determines the height of the curve, and γ is associated with the average age of mothers (Chandola et al. 1999).Th e density function for beta distribution is expressed as  ,wherein α and β represent distribution parameters and Β(α,β) is the beta func-tion.Neither the gamma distribution nor the Hadwiger distribution gave any adequate approximation for the observed years (Figures 2 and 3). 020406080100120140Gamma Observed Hadwiger BetaFigure 2. Age-specifi c fertility rates, 2008.Th e distribution that fi tted best for describing the curves under observation was the beta distribution.    Fertility modelling  |  111020406080100120140Gamma Observed Hadwiger BetaFigure 3. Age-specifi c fertility rates, 2008.As the next step, the parameters of the selected distribution are assessed for the fertility curves all of the observed years. 01234591 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09YearAlfa BetaFigure 4. Dynamics of distribution parameters.Th e change in the distribution parameters α and β is substantially linear. Th e value of the fi rst parameter is growing (Figure 4), the value of the parameter β decreases. 112  |  Mare VähiSo as to use the results obtained by fi tting the distributions to forecast the fertility in the following years, the possible future values of the parameters are assessed. Th e results are shown in Figures 5 and 6. 22,22,42,62,833,23,491 94 97 00 03 06 09 12 15 18 21YearAlphaAlpha Forecast Lower confidence limit Upper confidence limitFigure 5. The change of parameter alpha.Aft er that, the probable future values of the parameters for the following periods are determined. Th e values of the parameters and the predicted values with their confi dence level (confi dence level 0.95) are shown in Figures 5 and 6. Th e linear trend, of course, calls for caution when making long-term predic-tions. Th e said change will probably not last long.22,533,544,5591 94 97 00 03 06 09 12 15 18 21YearBetaBeta Forecast Lower confidence limit Upper confidence limitFigure 6. The change of parameter beta.    Fertility modelling  |  113By using the values of the parameters, the fertility curves for the coming years can be modelled. Th e fertility curve modelled for the year 2021 is been provided in Figure 7. 0,00,51,01,52,015 20 25 30 35 40 45 50RateAge2009 2021Figure 7. Age-specifi c fertility rate, 2009, 2021.It shows that the average age of mothers will probably become even higher and, in 2021, it might reach 32 years.In the following period, the age of mothers raises, but the shape of the curve remains comparable to that of the previous period. Similarly to other European countries, the average age of mothers is growing in Estonia, too. At present, the average age of the mother at childbirth is 29.9 years, which is by no means the highest in Europe. Th e average age of mothers in Europe is 30.5 years; the oldest mothers are in Austria and Italy where the average age of mothers is 31.8 and 31.7, respectively.REFERENCES1. Chandola T., Coleman D.A., Hirons R.W. (1999). Resent European fertility pat-terns: fitting curves to “distorted” distributions. Population Sudies, 53, 317–329.2. Gilje E. (1969). Fitting curves to age-specific fertility rates: Some examples. Sta-tistical Review of National Central Burleau of Statistics of Sweden, 7, 118–134.3. Hoem J.M., Madsen D., Nielsen J.L., Ohlsen E.-M., Hansen H.O., Renner-malm B. (1981). Experiments in modelling recent Danish fertility curves. Demography, 18 (2) , 231–244.4. Peristera P., Kostaki A. (2008). Modeling fertility in modern populations. Demographic Research, 16 (6), 141–194.114  |  Mare Vähi5. Rueda-Sabater C., Alvarez-Esteban P.C. (2008). The analysis of age-specific fertility patterns via logistic models. Journal of Applied Statistics, 35 (9), 1055–1070.Address for correspondence:Mare VähiInstitute of Mathematics and StatisticsFaculty of Science and TechnologyUniversity of Tartu, Tartu, EstoniaJ. Liivi 2–516, 50409 Tartu, EstoniaE-mail: mare.vahi@ut.ee

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Fertility modelling

In order to describe age-specific fertility curves, different theoretical approaches are employed. By modelling a time-variant fertility curve, it is possible to forecast reproductive behaviour of Estonian women in the future. The approach provided here is a novel way to forecast fertility.</jats:p

Mare Vähi

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Fertility modelling

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