Interpolation and forecasting of population census data

Abstract

Since population censuses are not annually implemented, population estimates are needed for the intercensal period. This paper describes simultaneous implementations of the temporal interpolation and forecasting of the population census data, aggregated by age and period. Since age equals period minus cohort, age-period-cohort decomposition suffers from the identification problem. In order to overcome this problem, the Bayesian cohort (BC) model is applied. The efficacy of the BC model for temporal interpolation is examined in comparison with official Japanese population estimates. Empirical results suggest that the BC model is expected to work well in temporal interpolation. Regarding the age-period-cohort decomposition of the Japanese census data, it is shown that the cohort effect is the largest while the other two effects are very small but not negligible. With regard to the forecasting of the Japanese population, the official population forecast considerably outperforms the BC forecast in most forecast horizons. However, the pace of increase in root mean square error for longer-term forecasting is larger in the official population forecast than in the BC forecasts. As a result, a variant of the BC forecast is best for 10-year forecast.

Appendix: Derivation of the Bayesian cohort model

In this Appendix are further details of the derivation of the BC model. I can rewrite the model (1) with the constraints (2) in vector and matrix notation as

$$ y = X\beta + \varepsilon $$

(5)

where $y = (Y_{11} ,Y_{21} , \ldots ,Y_{IJ} )',$ and$\varepsilon = (\varepsilon_{11} ,\varepsilon_{21} , \ldots ,\varepsilon_{IJ} )^{\prime}.$ X is an appropriate IJ × (I + J + K − 3) design matrix expressing the location of the effect parameters, A _i , P _j , and C _k , as shown in Table 2. The prime (′) denotes the transpose of a vector or a matrix. The likelihood of the model (5) is given by

$$ f(y|\beta ,\sigma^{2} ) = \,(2\pi \sigma^{2} )^{{{\frac{ - IJ}{2}}}} \exp \left\{ { - {\frac{1}{{2\sigma^{2} }}}(y - X\beta )'(y - X\beta )} \right\}. $$

(6)

In order to explicitly describe the constraint (3), Nakamura assumes that the parameter vector β has the prior distribution defined by

$$ \begin{aligned} \pi \left( {\beta |\sigma_{A}^{2} ,\sigma_{P}^{2} ,\sigma_{C}^{2} ,\sigma^{2} } \right) = & \left( {2\pi \sigma^{2} } \right)^{{ - {\frac{I + J + K - 3}{2}}}} |D^{\prime}\,\Upsigma^{ - 1} D|^{{\frac{1}{2}}} \\ & \times \exp \left\{ { - {\frac{1}{{2\sigma^{2} }}}\beta^{\prime}\,D^{\prime}\,\Upsigma^{ - 1} D\beta } \right\} \to \max , \\ \end{aligned} $$

(7)

where D is an (I + J + K − 3) × (I + J + K − 3) matrix expressing the first-order differences in the parameters, and $\Upsigma = {\text{diag}}\left\{ {\sigma_{A}^{2} , \ldots ,\sigma_{A}^{2} ,\sigma_{P}^{2} , \ldots ,\sigma_{P}^{2} ,\sigma_{C}^{2} , \ldots ,\sigma_{C}^{2} } \right\}$. In the case of I = 3 and J = 4, D is represented as

$$ D = \left( {\begin{array}{*{20}c} 1 & { - 1} & {} & {} & {} & {} & {} & {} & {} & {} \\ 1 & 2 & {} & {} & {} & {} & {} & {} & {} & {} \\ {} & {} & 1 & { - 1} & 0 & {} & {} & {} & {} & {} \\ {} & {} & 0 & 1 & { - 1} & {} & {} & {} & {} & {} \\ {} & {} & 1 & 2 & 2 & {} & {} & {} & {} & {} \\ {} & {} & {} & {} & {} & 1 & { - 1} & 0 & 0 & 0 \\ {} & {} & {} & {} & {} & 0 & 1 & { - 1} & 0 & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 1 & { - 1} & 0 \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & 1 & { - 1} \\ {} & {} & {} & {} & {} & 1 & 1 & 1 & 1 & 2 \\ \end{array} } \right) $$

If one specifies the values of σ ² _A , σ ² _P , and σ ² _C , it is reasonable to estimate β by the mode of the posterior density proportional to $f\left( {y|\beta ,\sigma^{2} } \right) \cdot \pi \left( {\beta |\sigma_{A}^{2} ,\sigma_{P}^{2} ,\sigma_{C}^{2} ,\sigma^{2} } \right)$ obtained by combining Eqs. (6) and (7).

The remaining problem is how to determine the values of σ ² _A , σ ² _P and σ ² _C . As a criterion for the determination of σ ² _A , σ ² _P and σ ² _C , Nakamura adopts the ABIC which is defined by

$$ \begin{aligned} {\text{ABIC}} = & - 2\log \left( {{\text{marginal}}\;{\text{likelihood}}} \right) + 2h \\ = & - 2\,\ln \int {f\left( {y|\beta ,\sigma^{2} } \right)} \cdot \pi \left( {\beta |\sigma_{A}^{2} ,\sigma_{P}^{2} ,\sigma_{C}^{2} ,\sigma^{2} } \right)\,d\beta + 2h ,\\ \end{aligned} $$

where h is the number of hyperparameters. In the BC model, the value of ABIC is evaluated approximately by

$$ \begin{gathered} {\text{ABIC}} = \left( {IJ} \right)\ln \hat{\sigma }^{2} - \ln |D^{\prime}\,\Upsigma^{ - 1} D| + \ln |X^{\prime}_{{}} \,X + D^{\prime}\,\Upsigma^{ - 1} D| + 2h \hfill \\ \hat{\sigma }^{2} = \left\{ {(y - X\hat{\beta })^{\prime}(y - X\hat{\beta }) + \hat{\beta }^{\prime}_{{}} D^{\prime}\,\Upsigma^{ - 1} D\hat{\beta }} \right\}/IJ \hfill \\ \hat{\beta } = (Q^{\prime}R^{ - 1} Q)^{ - 1} Q^{\prime}R^{ - 1} u, \hfill \\ \end{gathered} $$

where $u = \left( \begin{gathered} y \hfill \\ 0 \hfill \\ \end{gathered} \right),\,\,Q = \left( \begin{gathered} I\,\,\,\,X \hfill \\ 0\,\,\,\,\,D \hfill \\ \end{gathered} \right),\,\,R = \left( \begin{gathered} I\,\,\,\,0 \hfill \\ 0\,\,\,\,\Upsigma \hfill \\ \end{gathered} \right).$

For further details of the derivation of the estimate $\hat{\beta }$, its standard error, and ABIC, see Fukuda (2006).