Reconciling Forecasts of Infant Mortality Rates at National and Sub-National Levels: Grouped Time-Series Methods

Abstract

Mortality rates are often disaggregated by different attributes, such as sex, state, education, religion, or ethnicity. Forecasting mortality rates at the national and sub-national levels plays an important role in making social policies associated with the national and sub-national levels. However, base forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider the problem of reconciling mortality rate forecasts from the viewpoint of grouped time-series forecasting methods (Hyndman et al. in, Comput Stat Data Anal 55(9):2579–2589, 2011). A bottom-up method and an optimal combination method are applied to produce point forecasts of infant mortality rates that are aggregated appropriately across the different levels of a hierarchy. We extend these two methods by considering the reconciliation of interval forecasts through a bootstrap procedure. Using the regional infant mortality rates in Australia, we investigate the one-step-ahead to 20-step-ahead point and interval forecast accuracies among the independent and these two grouped time-series forecasting methods. The proposed methods are shown to be useful for reconciling point and interval forecasts of demographic rates at the national and sub-national levels, and would be beneficial for government policy decisions regarding the allocations of current and future resources at both the national and sub-national levels.

Appendix: Maximum Entropy Bootstrap Algorithm

An overview of the maximum entropy bootstrap algorithm is provided for generating a random realization of a univariate time series $x_t$. Consult Vinod (2004) for more details and an example. In the maximum entropy bootstrap algorithm, the following steps are followed:

• 1. Sort the original data in increasing order to create order statistics $x_{(t)}$ and store the ordering index vector.

• 2. Compute intermediate points $z_t = \frac{x_{(t)}+x_{(t+1)}}{2}$ for $t=1,\dots ,n-1$ from the order statistics.

• 3. Compute the trimmed mean, denoted by $m_{\text {trim}}$ of deviations $x_t - x_{t-1}$ among our consecutive observations. Compute the lower limit for the left tail as $z_0 = x_{(1)} - m_{\text {trim}}$ and the upper limit for the right tail as $z_n = x_{(n)}+m_{\text {trim}}$. These limits become the limiting intermediate points.

• 4. Compute the mean of the maximum entropy density within each interval such that the “mean-preserving constraint” is satisfied. Interval means are denoted as $m_t$. The means for the first and last intervals have simpler formulas:

  $$\begin{aligned} \left\{ \begin{array}{ll} m_1=0.75 x_{(1)}+0.25 x_{(2)} \\ m_k = 0.25x_{(k-1)} + 0.5 x_{(k)} + 0.25 x_{(k+1)}, \qquad k=2,\dots ,n \\ m_n = 0.25 x_{(n-1)} + 0.75 x_{(n)} \end{array} \right. \end{aligned}.$$

• 5. Generate random numbers from Uniform[0, 1], and compute sample quantiles of the maximum entropy density at those points and sort them.

• 6. Re-order the sorted sample quantiles by using the ordering index of Step 1. This recovers the time-dependence relationships of the originally observed data.

• 7. Repeat Steps 2 to 6 several times.