Modeling and Forecasting of l₂ (Survivors at Age 2) Values for Male and Female Population of Bangladesh: Regression Model Approach

Md.Rafiqul Islam

INTRODUCTION

Modeling in Demography in the Asian region, in particular, in Bangladesh has hardly ever been used. For representing data in the up to date hi-tech era, statistical models are very sophisticated and pragmatic devices. The statistical model is very much significant and imperative in differentiating necessary and unnecessary characteristics in the midst of a variety of socio-economic and demographic phenomena. Modeling is in fact essentially an endeavor to find out the functional interaction and their vibrant behaviors surrounded by the various components, not only in demographic but also in socio-economic analysis. Last but not least, a model is very important for the estimation of population projections and estimations. Normally, one can depict a number of figures for the demographic parameters as well as socio-economic indicators but, in the perception of Bangladesh, very few of us comprehend which types of functional or mathematical shapes are more appropriate for the parameters and social indicators. In this study, l₂ is defined as the number of survivors at age 2 in the l_x function of a life table in life table analysis. l₂ is very important and needed for the linking to adult mortality to attain a complete l_x function or column of a life table that is employed in the application of Orphanhood method, Widowhood method and other indirect methods  for the estimation of demographic parameters of Bangladesh.

Ali (1994) found that the relationship of total separation rates and separation rates due to death, with their age variable and found a semi-log function of the type . In Islam et al (2003), it was reported that age distribution, age specific death rates (ASDRs) and the number of persons surviving at an exact age x (l_x) for male population of Bangladesh in 1991 follow a modified negative exponential model, 4^th degree polynomial model and 3^rd degree polynomial model, respectively. Islam (2005) observed that age structure, ASDRs and l_x for female population of Bangladesh follow a modified negative exponential model, 4^th degree polynomial model and biquadratic polynomial model respectively. It was set up that the values of a life table for the male population followed a four parameters 3rd degree polynomial model, i. e. cubic polynomial model (Islam, 2006).

Therefore, the fundamental aims and objectives of this study are as follows:

i) to build up time trend statistical models to l₂ values for male and female population of Bangladesh and,
ii) then to forecast these values employing these fitted statistical regression models for 2008-2031.

DATA AND METHODOLOGY

Data Sources

To fulfill the objectives mentioned above the secondary data on l₂ values for male and female population of Bangladesh have been taken from (Islam, 2003, 2006, 2007 and 2007). These have been utilized as raw materials in the current study that are shown in Table 1.

Data Smoothing

It is observed that there is some kind of unpredicted distortions in the data aggregate if it is placed on a graph paper. Therefore, before going to fit the models to this data, an adjustment is needed to relieve these unpredicted distortions. So, these are smoothed using the Package Minitab Release 12.1 by the latest smoothing technique “4253H, twice” (Velleman, 1980). Afterward, the smoothed data are used to fit statistical models and these are shown in Table 1.

Regression Model Fitting

Using the scattered plot of l₂ values for male and female population of Bangladesh, it appears that these are linearly distributed. Therefore, a statistical model, that is, a simple linear regression model is considered and the structure of the model is

where, t represents time (years); y_t represents l₂ values;  a₀, a₁ are unknown parameters and u is the stochastic disturbance term of the model.

Note that these models are fitted using the software STATISTICA.

Model Validation Technique

To test out the validity or legitimacy of these models, the CVPP, , is applied. The mathematical formulation for CVPP is given as

; where, n is the number of classes, k is the number of regressors in the model and the cross-validated R is the correlation between observed and predicted values of the dependent variables (Stevens, 1996). The shrinkage of the model is the positive value of ( - R2); where is CVPP and R² is the coefficient of determination of the fitted model. As well, 1-shrinkage is the stability of R² of the model. The estimated CVPP analogous to their R² and information on model fittings are summarized in Table 2. It is noted that CVPP was also employed by Islam (2003 and 2005), Islam et al (2003 and 2005) and Khan and Ali (2004) as the model justification method.

To find out the overall measure of significance level of the fitted models as well as the significance of R² , the F-test is employed in this information. The F-test is specified by
with (l-1, n-l) degrees of freedom (d.f.);
where l = the number of parameters is to be estimated in the fitted model, n is the number of cases and R2 is the coefficient of determination of the model (Gujarati, 1998).

Table 1 Observed, Predicted and Residual of l₂ Values for Male and Female Population of Bangladesh During 1961-2007

Table 2 Information on Model Fittings

Table 3 Forecasted l2 Values for Male and Female Population of Bangladesh During 2008-2031

Figure 1 Observed, Smoothed and Predicted l2 Values for Male Population of Bangladesh. X axis represents Year (Time) and Y axis represents l2 Values.

Figure 2 Observed, Smoothed and Predicted l2 Values for Female Population of Bangladesh. X axis represents Year (Time) and Y axis represents l2 Values.

Figure 3 Forecasted l2 Values for Male and Female Population of Bangladesh During 2008-2031. X axis represents Year (Time) and Y axis represents forecasted l2 Values.

RESULTS and DISCUSSION

The statistical models, that is, simple linear regression model is assumed to fit to l2 values for male and female population of Bangladesh and the fitted time trend models are in the following:

y_t=-5.2775 +0.00308t for male … (1)
t-stats (-12.2147) (14.1946)

y_t=-4.60916 +0.00275t for female … (2)
t-stats (-9.72565) (11.53224)

The information on model fittings and estimated CVPP, , analogous to their R² of these models is shown in Table 2. From this table it appears that the fitted models (1) - (2) are highly cross-validated and their shrinkages are 0.0173 and 0.0259 respectively. These imply that the fitted models (1) - (2) will be stable more than 95% and 93% respectively. Moreover, it is found that the parameters of the fitted models (1) - (2) are highly statistically significant with significant of variance explained. The stability for R2 of these models is more than 98% and 97% respectively.

The calculated values of F statistic for the models (1) - (2) are 201.53 with (1, 5) d.f. and 133.01 with (1, 5) d.f. respectively whereas the analogous tabulated values are only 16.3 for (1) - (2) models at 1% level of significance. Therefore, from these statistics it is seen that these models and their analogous R2 are highly statistically significant. Hence, the fits of these models are well.

It should be mentioned here for information that others models such as exponential, logistic, quadratic, cubic, biquadratic were also applied to fit model to these data but those are not fit well due to shrinkage and proportion of variance explained.
Thereafter, the forecasted values are estimated using these fitted time trend regression models that are presented in Table 3. It is found from the Table 3 that l2 values are increasing, i.e., upward trend due to time during the forecasted period 2008-2031.

CONCLUSION

In this study it is found that l₂ values for male and female population of Bangladesh follow a simple linear regression model. Then these are forecasted using these statistical models during 2008– 2031. These might be used as predicted l₂ values for male and female population of Bangladesh for 2008–2031 for further higher study as these may be used in the application of Orphanhood method, Widowhood method and other indirect techniques for the estimation of demographic parameters for the forecasted period 2008– 2031.

REFERENCES

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