Projection of Indian Population by Using Leslie Matrix with Changing Age Specific Mortality Rate, Age Specific Fertility Rate and Age Specific Marital Fertility Rate

Abstract

The importance of projection in national and state level planning and policy formulation is quite well recognized in any country which attempts at achieving sustainability in human development and improving the quality of life. Population projection exercises are basically a part of forecasting growth of human population in the future years over a time horizon. There are various means of extrapolating past trend of change in human population over the future years. These means are determined by the assumptions that are made on the determinants of population change such as time, pattern of changes in fertility, mortality, and migration and other associated factors. The success of population projection depends not only on the technique of projection but also on the proximity of the assumptions to reality so that changes in the future years get estimated with least possible errors. Projections, however, might not suffice when there are significant deviations from the assumption that prevailing conditions would continue unchanged in the future. Also, the projection might not be satisfactory due to failure to incorporate adequately the changes in the policy parameters, technological changes, changes in the migration pattern, etc. Forecasting attempts at overcoming these drawbacks by incorporating the elements of judgment in the projection exercise. Forecasting enjoys the advantage of being based upon one or more assumptions that are likely to be realized in the future years. Thus, forecasts give more realistic picture of the future.

Appendix

13.1.1 A.1.1 Exponential Smoothing

It is one of the most popular types of automatic forecasting algorithms, which is used for the determination of an appropriate time series model, estimate of the parameters and compute the forecasts. Exponential smoothing methods were originally classified by Pegels’ (1969) taxonomy. This was later extended by Gardner (1985), modified by Hyndman et al. (2002), and extended again by Taylor (2003), giving a total of 15 methods seen in Table A.1.

Table A.1 The fifteen exponential smoothing methods

Table A.2 Formula for recursive calculations and point forecasts

13.1.2 A.1.2 Point Forecasts for All Methods

We denote the observed time series by ${y_1}\ldots {y_n}$. A forecast of ${y_{t+h }}$ based on all of the data up to time t is denoted by ${{\hat{y}}_{{\mathrm{t}+\mathrm{h}|\mathrm{t}}}}$. To illustrate the method, we give the point forecasts and updating equations for method (A, A), the Holt–Winters’ additive method:

$$ \mathrm{Level}:\quad\ {l_t}=\alpha \left( {{y_t}-{s_{t-m }}} \right)+\left( {1-\alpha } \right)({l_{t-1 }}+{b_{t-1 }}) $$

$$ \mathrm{Growth}:\quad {b_t}=\dot{\beta}\left( {{l_t}-{l_{t-1 }}} \right)+\left( {1-\dot{\beta}} \right){b_{t-1 }} $$

$$ \mathrm{Seasonal}:\quad\ {s_t}=\gamma \left( {{y_t}-{l_{t-1 }}-{b_{t-1 }}} \right)+\left( {1-\gamma } \right){s_{t-m }} $$

$$ \mathrm{Forecast}:\quad {{\hat{y}}_{{\mathrm{t}+\mathrm{h}|\mathrm{t}}}}={l_t} + {b_t}h+{s_{{t-m+h_m^{+}}}} $$

where m is the length of seasonality (e.g., the number of months or quarters in a year), ${l_t}$ represents the level of the series, ${b_t}$ denotes the growth, ${s_t}$ is the seasonal component, ${{\hat{y}}_{{\mathrm{t}+\mathrm{h}|\mathrm{t}}}}$ is the forecast for h periods ahead, and $h_m^{+} = \left[ {\left( {\mathrm{h}-1} \right)\ \mod\ \mathrm{m}} \right]+1$. To use this mentioned method, we need values for the initial states ${l_0}$, ${b_0}$ and ${s_{1-m }},\ldots,{s_0}$, and for the smoothing parameters α, $\dot{\beta}$ and γ. All of these will be estimated from the observed data (Table A.2).

The triplet (E, T, S), which refers to the three components: Error, Trend, and Seasonality, is added to the models for discrimination among the additive and multiplicative errors. So the model ETS (A, A, N) has additive errors, additive trend and no seasonality—in other words, this is Holt’s linear method with additive errors. Similarly, ETS (M, M_d, M) refers to a model with multiplicative errors, a damped multiplicative trend and multiplicative seasonality. The notation ETS (•, •, •) helps in remembering the order in which the components are specified.

13.1.3 A.1.3 Models for All Exponential Smoothing Methods

The general model involves a state vector ${{\boldsymbol{x}}_t}=({l_t},{b_t},{s_t},{s_{t-1 }},\ldots,{s_{t-m+1 }})$ and state space equations of the form

$$ {y_t}=w\left( {{{\boldsymbol{x}}_{t-1 }}} \right)+r\left( {{{\boldsymbol{x}}_{t-1 }}} \right){\varepsilon_t}$$

(13.17)

$$ {{\boldsymbol{x}}_t}=f\left( {\ {{\boldsymbol{x}}_{t-1 }}} \right)+g\left( {\ {{\boldsymbol{x}}_{t-1 }}} \right){\varepsilon_t} $$

(13.18)

where {${\varepsilon_t}$} is a Gaussian white noise process with mean zero and variance σ² and ${\mu_t}=w\left( {\ {{\boldsymbol{x}}_{t-1 }}} \right)$. The model with additive errors has $r\left( {{{\boldsymbol{x}}_{t-1 }}} \right)=1$, so that ${y_t}={\mu_t}+{\varepsilon_t}$. The model with multiplicative error has $r\left( {{{\boldsymbol{x}}_{t-1 }}} \right)={\mu_t}$, so that ${y_t}={\mu_t}(1+{\varepsilon_t})$. Thus, ${\varepsilon_t}=({y_t} - {\mu_t})/{\mu_t}$ is the relative error for the multiplicative model.

All of the methods in Table 13.8 can be written in the form (13.17) and (13.18). The specific form for each model is given in Hyndman et al. (2008).

13.1.4 A.1.4 Estimation

In order to use these models for forecasting, we need to know the values of ${x_0}$ and the parameters α, β, γ, and φ. It is easy to compute the likelihood of the above innovations state space model, and so obtain maximum likelihood estimates. Ord et al. (1997) show that

$$ {L^{*}}\left( {\boldsymbol{\theta}, {{\boldsymbol{x}}_0}} \right)=n\log (\mathop{\sum}\limits_{t=1}^n{\varepsilon_t}^2)+2\mathop{\sum}\limits_{t=1}^n\log |r\left( {{{\boldsymbol{x}}_{t-1 }}} \right)| $$

is equal to twice the negative logarithm of the likelihood function (with constant terms eliminated), conditional on the parameters θ = (α, β, γ, φ) and the initial states ${x_0}=({l_0},{b_0},{s_0},{s_{-1 }},\ldots,{s_{-m+1 }})$, where n is the number of observations.