Abstract Projection of future population for a country by age and - - PDF document

1 | P a g e

Probabilistic Approach of Population Projection for India and States

Anurag Verma1*, Abhinav Singh2

, P.S. Pundir2

• 1. Dept. of Community Medicine, IMS, BHU, Varanasi, India
• 2. Dept. of Statistics, University of Allahabad, Allahabad, India

*presenting author, e-mail:-imsbhuanurag@gmail.com

Abstract

Projection of future population for a country by age and sex, are widely used for policy development, planning and research. They are mostly done deterministically, but there is a widespread need for probabilistic projections. In this paper we propose a Bayesian method for probabilistic population projections for India. The total fertility rate (TFR) and life expectancies at birth are projected probabilistically using Gompertz and logistic growth models respectively under Bayesian paradigm. The estimates obtained from proposed two models combined using cohort-component method to obtain age-specific projection of the population by sex. The analysis has been made using Markov Chain Monte Carlo (MCMC) technique with the software OpenBUGS. Convergence diagnostics techniques available with the OpenBUGS software have been applied to ensure the convergence of the chains necessary for implementation of MCMC. The method is illustrated by making 40- year projection using Indian data for the period 1971-2011. The study will provide probabilistic point estimates of parameter as well as the projection along with highest posterior density (HPD) interval, which is derived from population number and vital events, includes age specific death rates, life expectancies, age specific birth rate, total fertility rates and dependency ratio. Keywords: Demography, Probabilistic Population Projection, Bayesian Approach, Posterior Distribution

Introduction

Population Growth has become one of the most important problems in the world [1]. The idea

• f the future population is achieved with the help of projection. The demand of precise

projected figures is always requisite for government personals, actuaries, for their social,

2 | P a g e economic planning purposes. The size of the population and the growth of a country directly affect the state of the economy, politics, culture, education and the environment, etc. In this country, and determine the cost of exploring natural sources no one wants to wait until these resources are depleted because of the population explosion. Therefore, the study of population projection has been started earlier [2-4]. Currently, there are two main approaches in statistics, such as the Frequentist and the Bayesian approach for data analysis [5]. The utilization of Bayesian approach in the field of data analysis is moderately new and has discovered mass support throughout the previous two decades to persons belonging to various disciplines. Probably the main reason behind the growing support is its flexibility and generality that allows it to deal with the complex

• situation. In addition, the Bayesian method is typically preferred by the classical method in

estimate parameters causing intractable from of the likelihood function [5]. Difficult situation can be handled by BUGS (Bayesian analysis using Gibbs Sampling) software for its flexibility and overall approach [6]. This study is based on a Bayesian way of data analysis. Bayesian method, uncertainty in model choice is incorporated through averaging techniques. Here the resulting predictive distributions from Bayesian forecasting models have two main advantages over those obtained using more traditional stochastic models. First, uncertainties in the data, the model parameters and model choice are explicitly represented using probabilistic distributions. As a result, more realistic probabilistic population forecasts are

• btained. Second, Bayesian models formally allow the incorporation of expert opinion

including uncertainty into the forecasts [7, 8]. Most population projection are currently done deterministically, using the cohort component method [9, 10], this is an age and sex-structured version of the basic demographic identity that the population of a country at the next time point is equal to the population at the current time point, plus the number of births minus the number of deaths, plus the number of immigrants minus the number of emmigrants. It was formulated in matrix form by Leslie [11]. Population Projection are currently produced by many organization, including national and local governments and private companies. The main organizations that have produced population projection for all states including India is Registrar General of India. In India’s current method (RGI, 2006) [12] does not yield an assessment of uncertainty about future population.

3 | P a g e Standard population projection methods are deterministic, meaning that they yield a single projected value for each quantity of interest. However, Probabilistic projection that gives a probability distribution of each quantity of interest and hence convey uncertainty about the projections are widely desired [13-14]. In the recent past, researcher developed alternative methods which allowed for probabilistic population projection, aimed for probabilistic interpretation of each demographic factor of

• interest. Alho and Spencer (1985), Alho (1990) ,Cohen (1986, 1988), Pflanumer (1988), Lee

(1992) and Lee and Tuljapurkar (1994),Allho (1999),Keilman (2002) show a probabilistic population projection. The comparison of deterministic and probabilistic method can be found by Lee(1998), Alho& Spencer (2006) and Stillwell&Clarke (2011). In this study expand on the work of Rahul(2007) up on is suggestion to integrate statistical and demographic methodology in performing age-specific population projection. The aim of the study is to improve current methodology in population projection for making probabilistic population projection using cohort component method under Bayesian approach for India and States. The total fertility rate and female and male life expectancies at birth are projected probabilistically using Bayesian models estimated via Markov Chain Monte Carlo under WinBUGS software using Indian population data. These are then converted to age- specific rates and combined with a Cohort Component Projection model. This yields Probabilistic projections of any population quantity of interest, the method is illustrated for four Indian state of different demographic stages, continents and sizes. In India’s current projection method dose not yield an assessment of uncertainty about future population quantities. It is somewhat subjective because the model used have been selected by the analyst from a small number of predetermined possibilities rather than estimated from the data. It is also somewhat rigid in that the set of model used is small and may not cover a full range of realistic future possibilities. To address these issues, we will develop a Bayesian probabilistic population projection method. This involves building Bayesian models to project the fertility and mortality rate, each of which produces a large number of possible future trajectories from the posterior predictive distribution. These are then input to the cohort component projection method to provide a posterior predictive distribution of any future population quantity of interest.

4 | P a g e

Methodology

Cohort Component Projection Method

The procedure for making cohort-component population projections was developed by Whelpton in the 1930s. It is uses the components of demographic change to project population growth. The technique projects the population by age groups, in addition to other demographic attributes such as sex and ethnicity. This projection method is based on the components of demographic change including births, deaths, and migration. It can be thought

• f as an elaboration of the ideas encapsulated in the demographic balancing equation:

𝑄𝑝𝑞𝑣𝑚𝑏𝑢𝑗𝑝𝑜𝑢+𝑜 = 𝑄𝑝𝑞𝑣𝑚𝑏𝑢𝑗𝑝𝑜𝑢 + 𝐶𝑗𝑠𝑢𝑖𝑡𝑢 − 𝐸𝑓𝑏𝑢𝑖𝑢 + 𝐽𝑛𝑛𝑗𝑕𝑠𝑏𝑢𝑗𝑝𝑜𝑢 − 𝐹𝑛𝑗𝑕𝑠𝑏𝑢𝑗𝑝𝑜𝑢 (1)

where, Populationt is the population at time t, Birthst and Deatht are number of births and deaths occurring between t and t+n. Immigrationt and Emigrationt are the number of immigrants and of emigrants from the country during the period t to t+n. This equation reminds us that there are only two possible ways of joining a population: one can be born into it or one can migrate into it. Similarly, the only ways to leave a population are to emigrate or to die. Cohort-component projections extend this concept to individual age cohorts. They make use of the fact that every year of time that passes, every member of a population becomes a year older. Thus, after 5 years the survivors of the cohort aged 0-4 years at some baseline date will be aged 5-9 years, 5 years after that they will age 10-14 years, and so on. The Age-specific fertility (the ability of an individual to give a livebirth) rates are required to project the number of births in future fertility projections, which are made by simulate a large number of trajectories of future values of the total fertility rate (TFR) and convert them to age-specific fertility rates using model fertility schedules. Next, the Age and sex-specific death rates required to project the total deaths in future mortality projections, which are made by simulate an equal number of trajectories of life expectancy at birth for females and males and convert them to age-specific mortality rates using a model life table. In this study, to simulate future values of TFR by using Gompertz model and to simulate future values of life expectancy at birth separately for both the sex by using Logistic model under the Bayesian approach.

5 | P a g e

Bayesian Inference

In Bayesian inference, uncertainty about the parameters 𝜄 of a statistical model is described by its posterior probability distribution given observed data 𝑦𝑗 = 𝑦1, 𝑦2, … . . , 𝑦𝑗 . The probability density function of 𝑦𝑗 is obtained by using Bayes Theorem: 𝑔 𝜄 𝑦𝑗 =

𝑔(𝑦𝑗|𝜄)𝑔(𝜄) 𝑔(𝑦𝑗)

, (a) where𝑔(𝑦𝑗|𝜄) is the likelihood function and is defined by the model, 𝑔(𝜄) is the prior distribution for 𝜄 and 𝑔(𝑦𝑗) is a normalizing constant. The prior distribution 𝑔(𝜄) specifies the uncertainty about 𝜄 prior to observing any data. Forecasting or prediction is particularly natural in a Bayesian framework. Uncertainty about next N future values of 𝑦𝑜(for n=i+1,…..,i+N) is described by the joint predictive probability distribution 𝑔(𝑦𝑗+1, … … , 𝑦𝑗+𝑂|𝑦𝑗) = 𝑔(𝜄 |𝑦𝑗) 𝑔(𝑦𝑗+𝑂|𝑦𝑗+𝑂−1,

𝑂 𝑘=1

𝜄)𝑒𝜄. (b) Note that the product term represent the joint predictive distribution in the case that parameter 𝜄 is known. The Bayesian predictive distribution simply averages or integrates this with respect to the posterior probability distribution for 𝜄. Hence, uncertainty about 𝜄 in light of the observed data is fully integrated. In a Bayesian analysis we obtain forecasts and associated measures of uncertainty by calculating marginal probability distributions for quantities of interest by integrating the posterior distribution in (a) or the predictive distribution in (b). Performing these integrations analytically is typically not possible for realistically complex models such as those described

• above. Historically, this has been prevented demographers and others from taking advantages
• f Bayesian methods for statistical inference. Recent developments in Bayesian computation

have focused on Markov chain Monte Carlo (MCMC) generation of samples from distributions such as (a) and (b); see Gelman et al.(2004) for details. Once a samples has been

• btained from a joint distribution, then a sample from a distribution of any component or

function of components is readily available. To generate samples from the posterior and predictive distribution in this study, used an MCMC sampling approach implemented using the WinBUGS software.

Probabilistic Projection of Fertility

6 | P a g e Let Yi to denote TFR of India in the year ti (i=1,…..,n), where i refers to TFR data in the ith year in a country and its initial value is taken to be 1. The four parameter Gompertz model use for the projection of Total fertility Rate may be described as follows. Assume general regression equation: 𝑍𝑗 = 𝜈𝑗 + 𝜁𝑗 Where 𝑍𝑗 be the population size, in the year 𝑢𝑗 has been assumed to follow normal distribution with respective mean µi and common precision (=1/variance)T. Here , µi is the deterministic part and 𝜗𝑗 is the disturbance part. Therefore, the disturbance part 𝜗𝑗~iid N(0,𝜐). For the implementation of four parameter Gompertz model, we assume deterministic part as: 𝜈𝑗 = 𝜄1 + 𝜄2𝑓−𝑓

(−𝜄3−𝜄4 𝑢𝑗−𝑢 𝜏 )

Where, 𝜄1is the lower asymptote and in our study this parameter is fixed at 1.8, 𝜄2 is the upper asymptote, 𝜄3 is the parameter that determines the shape of the Gompertz curve, 𝜄4 is the rate at which the fertility increases. To adopt Bayesian analysis, we need to provide prior distributions to all the parameters present in the model, 𝜄2, 𝜄3, 𝜄4 and 𝜐. We prefer to assign non-informative priors for all of them as N(0,0.01), (variance=1/0.001) prior has been assigned to all of the parameters 𝜄2, 𝜄3, 𝜄4 and Gamma(0.00001,0.00001) to the parameter 𝜐.

Probabilistic Projection of Mortality

Let 𝑍𝑗𝑘 denote the Life expectancy at birth of a country in the year 𝑢𝑗(i=1,2,3,…..,n), where i represents the time and j represents the sex agin its value is taken as 1. The four parameter logistic model is used for the projection of Life expectancy at birth. Let us assume the general regression equation as: 𝑍𝑗𝑘 = 𝜈𝑗𝑘 + 𝜗𝑗𝑘 Where, 𝑍𝑗𝑘 be the life expectancy at birth for males and females of a country in the year 𝑢𝑗(i=1,2,…,n), where i represents time and j represent the sex. 𝜈𝑗𝑘 is the deterministic part and 𝜗𝑗𝑘 assumes to be error term which follows iid N(0,𝜐𝑘), where 𝜐𝑘 is the precision. The deterministic part of our logistic model is:

7 | P a g e 𝜈𝑗𝑘 = 𝜄1𝑘 + 𝜄2𝑘 (1 + 𝑓𝑓2𝑘𝑓

𝜄3𝑘

𝑢𝑗−𝑢 𝜏𝑢 )

+ 𝜄4𝑘; 𝑘 = 1,2 Where 𝜄1𝑘is the lower asymptote, 𝜄2𝑘 ,𝜄3𝑘 , 𝜄4𝑘is the………. In this model, the life-expectancy at birth 𝑍𝑗𝑘 in the year 𝑢𝑗 has been assumed to follow normal distribution with respective means 𝜈𝑗𝑘 and common precisions 𝜐𝑘. We need to assign priors for parameter. We prefer to assign non informative priors have been assigned to all the parameters of the model.Normal(0,0.01), (variance=1/0.001) prior has been assigned to the parameters 𝜄1𝑘, 𝜄2𝑘, 𝜄3𝑘,𝜄4𝑘 and Gamma(0.00001,0.00001) prior to the parameter 𝜐𝑘. Future migration is more difficult to project that fertility or mortality. Migration can be volatile since short-term changes in economic, social, or political factors often play an important role. In this study, we assumed that the population is closed, i.e no migration takes place, or even if it does, net effect is zero. As for the sex ration at births which divide the future number of newborn into male and female, the female to male ratio is set 100:105 it is based on biological literature, and it remains consistent from 2011 onward.

Data

In this section, illustrate projecting method with the data for India. These data represent a case in which the counts of all population components by five year of age and sex are

• available. The data used to produce projection represent the three year moving average from

period 1971-2013. The data on mortality and fertility rates were obtained from published statistical report of SRS. The India population obtain for Census 2001 and 2011, were 2011 census used as a baseline for prediction, was also obtained from the office of Registrar General India. To test and implement the proposed model in this study selected four states namely Gujarat, Kerala, Orissa and Uttar Pradesh besides the country as a whole. These states were selected considering their geographical and demographic diversities. While Uttar Pradesh is the largest populated state in India, Gujarat is an economically developed state but lacking far behind in terms human development indicators. Kerala ranks one while Gujarat ranks 7, Uttar Pradesh ranks 15 and Orissa ranks 17 among the states of India based on the report “Inequality-adjusted Human Development Index for India’s States 2011” published by

8 | P a g e UNDP India. In addition to the diversities, all these selected states are situated in four different corners of the country.

Results

A Win BUGS program was developed to make a Bayesian analysis for the data to provide projection of the TFR and Life-expectancy at birth for India and states. WinBUGS codes of the models are given in the appendix. During the implementation of the program, we have taken two chains to run for each programme.

Projection of Fertility

In this study, for fertility model, we monitored four nodes 𝜄2, 𝜄3, 𝜄4 and 𝜏 = 1/√𝜐 separately for different dataset. Here we present the various diagnostics for Indian TFR data based on 45000 iterations after burning 5000 initial iterations. The history plots of the sample values of four nodes 𝜄2, 𝜄3, 𝜄4 and 𝜏 against iterations the two chains of the model have been show in the Fig1(a-d). the mixing of the chains (in different colors) for all nodes in this model looks quite good giving us a confidence of the convergence of chains. Fig 2(a-d) presents autocorrelations for different lags for all the four nodes which decling trend with increase in lags. The value of R in the bgr diagnostic for the all nodes are also close to one as show in Fig3(a-d). the traces of the blue and green lines are stable and the red on has converged to one for all the four parameters monitored. Fig 4(a-d) shows smoothed curves of the posterior densities of the nodes. The appearance of all the curves is bell shaped indicating asymptotically normal.

9 | P a g e Fig1 (a-d) History Plots for parameters from two chains

(a) iteration 4999 20000 40000

theta[2] 4.06.08.0 12.0

(b) iteration 4999 20000 40000

theta[3]

• 0.04 -0.03 -0.02

(c) iteration 4999 20000 40000

theta[4]

• 10.00.010.0

20.0

(d) iteration 4999 20000 40000

sigma 0.050.10.150.2

10 | P a g e Fig2 (a-d) Autocorrelation plots for parameters from two chains Fig3 (a-d) bgr Plots for parameters from two chains

(a):theta[2] lag 50

• 1.0 0.0 1.0

(b):theta[3] lag 50

• 1.0 0.0 1.0

(c):theta[4] lag 50

• 1.0 0.0 1.0

(d):sigm a lag 50

• 1.0 0.0 1.0

(a):theta[2] start-iteration 5225 10000 20000

0.0 0.5 1.0

(b):theta[3] start-iteration 5225 10000 20000

0.0 0.5 1.0

(c):theta[4] start-iteration 5225 10000 20000

0.0 0.5 1.0

(d):sigm a start-iteration 5225 10000 20000

0.0 0.5 1.0

11 | P a g e

(c) theta[4]

• 10.0

0.0 10.0

0.0 0.1

(b) sigm a 0.05 0.1 0.15 0.2

0.020.0 40.0 Fig4 (a-d) Density Plots for parameters from two chains To obtain summary statistics for the estimates of the parameters of the model after burning 5000 initial samples, during these updates none of the diagnostics indicates any symptom of non-convergence of chain. Therefore, 45000 update where run after the initial 5000 update burn-in. Table 1, it provides that the summary statistics for the selected nodes for the Gompertz model fit for India and selected states. The value of the carrying or upper asymptote can be estimated as 4.04, 7.56, 8.07, 10.10, 12.27, TFR for Kerala, Orissa, India, Gujarat, and Uttar Pradesh respectively. The rate at which the fertility decaling is minimum 2% per year for Uttar Pradesh which still show the very slow rate of decaling in fertility and maximum decaling rate in fertility for Kerala, which show that the fasted rate of decaling in the fertility to reached in the replacement level of fertility.

(a) theta[2] 4.0 6.0 8.0 10.0

0.0 0.4

(b) theta[3]

• 0.04
• 0.035
• 0.03
• 0.025

0.0 200.0

12 | P a g e Table 1.Summary Statistics of the node of fertility model by India and states Nade (India /States) Mean SD MC error HPD region 2.50% Median 97.50% India theta[2] 8.07 0.67 0.005 6.80 8.06 9.43 theta[3]

• 0.03

0.00 0.000

• 0.03
• 0.03
• 0.02

theta[4] 5.53 3.27 0.024

• 0.90

5.53 11.87 Sigma 0.10 0.01 0.000 0.08 0.10 0.12 Gujarat theta[2] 10.10 0.82 0.006 8.56 10.07 11.79 theta[3]

• 0.03

0.00 0.000

• 0.03
• 0.03
• 0.02

theta[4]

• 1.40

2.77 0.020

• 6.86
• 1.39

3.99 Sigma 0.15 0.02 0.000 0.12 0.15 0.19 Kerala theta[2] 4.04 0.81 0.004 2.81 3.92 5.92 theta[3]

• 0.09

0.01 0.000

• 0.12
• 0.09
• 0.07

theta[4] 5.62 2.37 0.011 0.43 5.86 9.58 Sigma 0.09 0.01 0.000 0.07 0.09 0.12 Orissa theta[2] 7.56 0.72 0.005 6.20 7.54 9.03 theta[3]

• 0.03

0.00 0.000

• 0.03
• 0.03
• 0.03

theta[4] 4.94 3.22 0.021

• 1.38

4.94 11.26 Sigma 0.13 0.02 0.000 0.11 0.13 0.17 Uttar Pradesh theta[2] 12.27 0.67 0.006 10.97 12.27 13.60 theta[3]

• 0.02

0.00 0.000

• 0.02
• 0.02
• 0.02

theta[4] 4.31 3.17 0.028

• 1.85

4.28 10.55 Sigma 0.15 0.02 0.000 0.12 0.15 0.19 Figure 5 (a-d) shows that the graphical presentation of the fitting of the Gompertz model for different selected data set. The Fig. shows that the model provides a close fit to observed

• data. Black dotted show that the observe value of SRS TFR data and dark red line provide the

mean value of estimated and projected value of TFR and dotted blue line provide 95% HPD (highest Posterior Density) region or lower and upper credible interval.

13 | P a g e

• Figure5. Fitted, Projected and HPD region of the estimates on Gompertz model.

Projection of Mortality

In this study, for fertility model, we monitored four nodes 𝜄2, 𝜄3, 𝜄4 and 𝜏 = 1/√𝜐 separately for different dataset. To obtain summary statistics for the estimates of the parameters of the model after burning 5000 initial samples, during these updates none of the diagnostics indicates any symptom of non-convergence of chain. Therefore, 45000 update where run after the initial 5000 update burn-in. Table 2, it provides that the summary statistics for the selected nodes for male and female under the logistics model fit for India and selected states. The value of the carrying capacity

• r maximum Life- expectancy is estimated as 𝜄1 + 𝜄4 is 69, 67, 72, 65, 63 years for males in

India, Gujarat, Kerala, Orissa, and Uttar Pradesh respectively and for females it is 74, 70, 77, 68, 68, years for India, Gujarat, Kerala, Orissa, and Uttar Pradesh respectively.

14 | P a g e Table 2.1.Summary statistics of the node for the life-expectancy at birth for males for India and states Node (India /States) Mean SD MC error HPD region 2.50% Median 97.50% India theta[1] 35.39 5.98 0.04 24.71 35.09 47.78 theta[2]

• 2.17

0.22 0.00

• 2.65
• 2.15
• 1.80

theta[3]

• 1.52

0.25 0.00

• 2.09
• 1.49
• 1.14

theta[4] 34.47 5.06 0.04 23.70 34.82 43.21 Sigma 0.27 0.10 0.00 0.15 0.25 0.53 Gujarat theta[1] 31.43 6.23 0.04 20.85 30.94 44.81 theta[2]

• 3.11

0.43 0.00

• 4.10
• 3.06
• 2.39

theta[3]

• 2.27

0.46 0.00

• 3.36
• 2.20
• 1.58

theta[4] 36.15 5.58 0.03 23.87 36.70 45.32 Sigma 0.61 0.23 0.00 0.34 0.56 1.19 Kerala theta[1] 35.16 7.38 0.04 20.56 35.20 49.55 theta[2]

• 3.86

0.87 0.00

• 5.88
• 3.76
• 2.47

theta[3]

• 2.02

0.54 0.00

• 3.32
• 1.94
• 1.22

theta[4] 37.39 7.22 0.03 23.21 37.37 51.58 Sigma 0.83 0.32 0.00 0.46 0.76 1.63 Orrisha theta[1] 32.94 6.94 0.04 19.67 32.83 46.86 theta[2]

• 2.79

1.00 0.01

• 5.05
• 2.62
• 1.55

theta[3]

• 1.97

0.70 0.01

• 3.62
• 1.83
• 1.19

theta[4] 32.38 6.17 0.04 19.74 32.64 43.68 Sigma 1.13 0.52 0.00 0.58 1.01 2.44 Uttar Pradesh theta[1] 26.56 3.77 0.03 21.21 25.83 36.18 theta[2]

• 3.33

0.27 0.00

• 3.90
• 3.32
• 2.83

theta[3]

• 2.63

0.35 0.00

• 3.32
• 2.63
• 1.94

theta[4] 37.15 3.31 0.03 28.63 37.82 41.69 Sigma 0.28 0.11 0.00 0.15 0.26 0.57

15 | P a g e Table 2.2.Summary statistics of the node for the life-expectancy at birth for females for India and states Node (India /States) Mean SD MC error HPD region 2.50% Median 97.50% India theta[1] 39.32 6.43 0.05 27.25 39.14 52.46 theta[2]

• 2.14

0.33 0.00

• 2.91
• 2.09
• 1.63

theta[3]

• 1.71

0.33 0.00

• 2.51
• 1.65
• 1.24

theta[4] 34.65 5.08 0.04 24.01 34.90 43.83 Sigma 0.51 0.22 0.00 0.26 0.46 1.07 Gujarat theta[1] 34.65 6.99 0.04 21.55 34.49 48.67 theta[2]

• 3.66

1.31 0.01

• 6.89
• 3.40
• 2.15

theta[3]

• 2.66

0.99 0.01

• 5.01
• 2.47
• 1.59

theta[4] 36.07 6.40 0.04 22.87 36.37 47.63 Sigma 1.76 0.94 0.01 0.89 1.54 3.86 Kerala theta[1] 38.61 7.01 0.03 25.03 38.59 52.53 theta[2]

• 5.15

0.94 0.01

• 7.30
• 5.04
• 3.69

theta[3]

• 2.92

0.63 0.00

• 4.38
• 2.84
• 1.97

theta[4] 38.79 6.90 0.03 25.08 38.83 52.11 Sigma 0.93 0.36 0.00 0.51 0.85 1.85 Orrisha theta[1] 33.46 6.78 0.04 20.90 33.24 47.32 theta[2]

• 2.45

0.82 0.01

• 4.40
• 2.30
• 1.42

theta[3]

• 2.11

0.67 0.01

• 3.78
• 1.97
• 1.31

theta[4] 35.17 5.53 0.03 23.16 35.63 44.65 Sigma 1.19 0.58 0.00 0.59 1.05 2.62 Uttar Pradesh theta[1] 40.41 4.99 0.03 30.82 40.33 50.45 theta[2]

• 2.70

0.30 0.00

• 3.40
• 2.66
• 2.28

theta[3]

• 2.21

0.34 0.00

• 3.00
• 2.16
• 1.72

theta[4] 27.65 4.01 0.02 19.36 27.79 35.15 Sigma 0.45 0.20 0.00 0.23 0.40 0.97

16 | P a g e The graphical presentation of the models fitted and projected to both males and females are shows in figure 6 (a-b). The graphs show that the model provides a close fit to the observed

• data. Black dotted show that the observe value of SRS Life-expectancy at birth data and dark

red line provide the mean value of estimated and projected value of Life-expectancy at birth and dotted blue line provide 95% HPD (highest Posterior Density) region or lower and upper credible interval. The graph of the projections approaches to S-shape, indicating the stabilization of the life-expectancy at birth for males as well as females.

• Figure6. Fitted, Projected and HPD region of the estimates on Logistic model.

Probabilistic Population Projection

The result of projection the two components of population change- that is, samples from the posterior distributions of mortality and fertility rates are subsequently combined into a cohort component model. On the basis of the future projection of population growth components, the projected population with different characteristic for India and selected states from 2016 to 2061 has been presented in Table3. The population size decays in mean from 1259 million in 2016 to 1513 million in 2061 for India. The rapidly changing structure in the Indian population is reflected in Table 3. From these table show that the decreasing number of children and increasing number of the elderly in India as well as selected states. The old-age percentage with the limit 65 year rises from 5% in 2016 to 18% in year 2061. This shows that Indian population will move to enter in ageing

• country. In ageing population there is several impacts on population. If the retirement age

remains fixed and the life expectancy increases (by Fig. 6) there will be relatively more people claiming pension benefits and less people working and paying income taxes. These also need to increased government spending on health care and pensions. Now, this is a

17 | P a g e current time for Indian government and policy maker to take bold steps to reduction the impact of ageing population in Indian scenario.

Conclusion

The present study to prepare a procedure for Bayesian population projection using cohort method is perhaps first attempt of this type in our country. There have been other attempts for Bayesian population projection but they were not using cohort component methodology. (Rahul et. Al 2007) There is no dearth of projection exercise using cohort component methods but nearly all of them were deterministic type projections. (Dyson, T., 2004, Registrar General of India, 2006) As we discussed earlier, the uncertainty in the projections arises from the uncertainties in the component and base year population. We have made several hard and perhaps unrealistic assumptions regarding the estimates and projections of these components. In this study, we are unable to provide future value for the component of migration because of sparse data for India. To overcome this problem, we have used a strong assumption, and this is the major drawback of our study. In both fertility and mortality models, we have applied non-informative priors. These prior is not provide substantial information to posterior distribution and it is also a limitation of this study. However, they were necessary for the implementation of the Bayesian data analysis. We have not analyzed the accuracy and consistency of the past estimates of the components neither we attempted the accuracy them using other studies and surveys. Our main emphasis was to develop the methodology for population projection under Bayesian approach.

18 | P a g e

Reference

• 1. Ehrlich, P. R., & Holdren, J. P. (1971). Impact of population growth.
• 2. Keyfitz, N. (1972). On future population. Journal of the American Statistical

Association, 67(338), 347-363.

• 3. Lutz, W. (2013). The future population of the world: what can we assume today.

Routledge.

• 4. Repetto, R. (1987). Population, resources, environment: an uncertain future.

Population Bulletin, 42(2), 3.

• 5. Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2014). Bayesian data analysis

(Vol. 2). Boca Raton, FL, USA: Chapman & Hall/CRC.

• 6. Lunn, David, et al. The BUGS book: A practical introduction to Bayesian analysis.

CRC press, 2012.

• 7. Daponte, B. O., Kadane, J. B., &Wolfson, L. J. (1997). Bayesian demography:

projecting the Iraqi Kurdish population, 1977–1990. Journal of the American Statistical Association, 92(440), 1256-1267.

• 8. Abel, G. J., Bijak, J., Forster, J. J., Raymer, J., & Smith, P. W. (2010). Bayesian

population forecasts for England and Wales.

• 9. Registrar General, I. (2006). Population projections for India and states 2001–2026.

New Delhi: Office of the Registrar General and Census Commissioner.

• 10. Whelpton, P. K. (1936). An empirical method of calculating future population.

Journal of the American Statistical Association, 31(195), 457-473.

• 11. Leslie, P. H. (1945). On the use of matrices in certain population mathematics.

Biometrika, 33(3), 183-212.

• 12. Registrar General, I. (1997). Population Projections for India and States 1996-2016.

Government of India, New Delhi.

• 13. Alho, J. (1999). On probabilistic forecasts of population and their uses. Bulletin of the

International Statistical Institute, 52nd Session, Proceedings, 58.

• 14. Alho, J. M., & Spencer, B. D. (1985). Uncertain population forecasting. Journal of the

American Statistical Association, 80(390), 306-314.

• 15. Alho, J. M. (1990). Stochastic methods in population forecasting. International

Journal of forecasting, 6(4), 521-530.

• 16. Cohen, J. E. (1986). Population forecasts and confidence intervals for Sweden: A

comparison of model-based and empirical approaches. Demography, 23(1), 105-126.

19 | P a g e

• 17. Cohen, J. E. (1988). Erratum to: Population forecasts and confidence intervals for

Sweden: A comparison of model-based and empirical approaches. Demography, 25(2), 315-315.

• 18. Pflaumer, P. (1988). Confidence intervals for population projections based on Monte

Carlo methods. International Journal of Forecasting, 4(1), 135-142.

• 19. Lee, R. D. (1992). Stochastic demographic forecasting. International Journal of

Forecasting, 8(3), 315-327.

• 20. Lee, R. D., &Tuljapurkar, S. (1994). Stochastic population forecasts for the United

States: Beyond high, medium, and low. Journal of the American Statistical Association, 89(428), 1175-1189.

• 21. Keilman, N., Pham, D. Q., &Hetland, A. (2002). Why population forecasts should be

probabilistic–illustrated by the case of Norway. Demographic Research, 6(15), 409- 454.

• 22. Lee, R., &Tuljapurkar, S. (1998). Population Forecasting for Fiscal Planning: Issues

and Innovations.

• 23. Alho, J., & Spencer, B. (2006). Statistical demography and forecasting. Springer

Science & Business Media.

• 24. Stillwell, J., & Clarke, M. (Eds.). (2011). Population dynamics and projection

methods (Vol. 4). Springer Science & Business Media.

20 | P a g e Appendix BUGS CODE (WinBUGS) for Fertility model: model { for( i in 1 :90 ) { Y[i] ~ dnorm(mu[i], tau) Z[i] ~ dnorm(mu[i], tau) mu[i] <- theta[1] + theta[2]* exp(-exp(-theta[3]*(x[i]-theta[4]))) } theta[1]<-1.8 theta[2] ~ dnorm(0,0.01) theta[3] ~ dnorm(0,0.01) theta[4] ~ dnorm(0,0.01) tau ~ dgamma(0.001, 0.001) sigma <- 1 / sqrt(tau) } BUGS CODE was executed with 50,000 observations for simulation, a burn-in of 5,000, and refresh of 100, and it produced the posterior analysis. BUGS CODE (WinBUGS) for Mortality model (Use Separately for Male and Females : model { for( i in 1 : 90 ) { Y[i] ~ dnorm(mu[i], tau) Z[i] ~ dnorm(mu[i], tau) mu[i]<-theta[1]/(1+exp(theta[2])*exp(theta[3]*((x[i]- mean(x[]))/sd(x[]))))+theta[4] } for(i in 1:4){ theta[i]~dnorm(0,0.01) } tau ~ dgamma(0.001, 0.001) sigma <- 1 / sqrt(tau) } BUGS CODE was executed with 50,000 observations for simulation, a burn-in of 5,000, and refresh of 100, and it produced the posterior analysis.

21 | P a g e

• Table3. Projected Population Characteristics 2016-2061 for India and selected States.

(Population in Millions)

India 2016 2021 2026 2031 2036 2041 2046 2051 2056 2061 Total 1259 1334 1396 1444 1480 1508 1527 1535 1530 1513 Male 651 689 720 743 760 773 781 784 779 769 Female 608 645 676 701 720 735 746 752 750 744 Percentage 0-14 26.29 25.37 24.05 22.06 20.02 18.69 18.05 17.64 17.06 16.37 15-64 67.97 68.12 68.47 68.99 69.78 69.81 69.24 68.26 66.48 65.05 65 and above 5.74 6.52 7.49 8.95 10.19 11.5 12.71 14.11 16.46 18.59 15-49(Female Population) 55.37 54.22 53.09 52.73 52.45 50.27 47.61 46.32 45.55 44.95 Gujarat Total 63 66 69 71 73 74 75 75 76 77 Male 33 34 36 37 37 38 38 39 39 39 Female 30 32 33 34 35 36 36 37 37 38 Percentage 0-14 27.52 25.55 23.91 22.55 21.49 20.69 19.89 19.01 18.23 17.75 15-64 66.94 68.02 68.54 68.59 68.61 68.6 68.26 67.66 66.34 65.13 65 and above 5.54 6.43 7.55 8.85 9.9 10.71 11.86 13.33 15.43 17.11 15-49(Female Population) 53.59 53.24 52.81 51.9 51.08 49.88 48.84 48.34 46.64 45.23 Kerala Total 36 37 38 39 39 39 39 38 37 36 Male 18 18 19 19 19 19 19 19 18 18 Female 19 19 19 20 20 20 20 19 19 18 Percentage 0-14 20.85 19.73 18.48 17.43 16.76 16.29 15.91 15.56 15.29 15.15 15-64 69.7 69.14 69.06 68.03 67.03 65.64 64.14 62.95 61.95 62.03 65 and above 9.45 11.13 12.46 14.53 16.21 18.07 19.95 21.49 22.76 22.82 15-49(Female Population) 53.11 51.05 48.68 46.23 44.06 42.21 41.61 40.7 40.52 40.24 Orissa Total 43 45 47 47 48 48 48 48 47 46 Male 22 23 24 24 24 24 24 24 24 23 Female 21 22 23 23 24 24 24 23 23 22 Percentage 0-14 24.77 23.76 22.32 20.36 18.56 17.52 17.1 16.81 16.33 15.74 15-64 69.19 69.49 69.97 70.33 70.84 70.47 69.77 68.72 67.01 65.53 65 and above 6.04 6.76 7.71 9.3 10.6 12.01 13.12 14.46 16.67 18.73 15-49(Female Population) 56.49 55.17 53.73 53.13 52.07 49.5 46.57 45.56 44.99 44.55 Uttar Pradesh Total 220 241 261 279 294 308 322 333 340 345 Male 115 126 137 145 153 160 166 171 175 177 Female 104 115 125 133 141 149 155 161 165 168 Percentage 0-14 32.2 31.97 31.16 29.17 26.63 24.61 23.48 22.71 21.68 20.37 15-64 63.19 63.17 63.52 64.84 66.77 68.27 68.85 68.76 67.9 67.31 65 and above 4.62 4.87 5.32 5.99 6.6 7.12 7.67 8.53 10.41 12.32 15-49(Female Population) 53 52.06 51.83 52.71 54.03 53.24 51.37 50.94 50.76 50.71