Dynamic Mortality Forecasting via Mixed-Frequency State-Space Models

TL;DR

The paper develops a mixed-frequency Lee–Carter state-space model (MF-SS) that jointly models monthly deaths and annual mortality rates through a shared latent monthly factor, estimated with EM and Kalman filtering. It demonstrates that aggregating monthly forecasts often yields more accurate annual mortality projections than direct annual forecasts, and that intra-year nowcasting improves substantially when monthly data are incorporated. Temporal forecast reconciliation benefits independently estimated monthly and annual LC models but adds little to MF-SS forecasts, underscoring the advantage of cross-frequency pooling in MF-SS. Overall, MF-SS provides timely, probabilistic mortality forecasts with robust intra-year updating, while reconciliation remains a valuable tool for alternative modelling approaches.

Abstract

High-frequency death counts are now widely available and contain timely information about intra-year mortality dynamics, but most stochastic mortality models are still estimated on annual data and therefore update only when annual totals are released. We propose a mixed-frequency state-space (MF--SS) extension of the Lee--Carter framework that jointly uses annual mortality rates and monthly death counts. The two series are linked through a shared latent monthly mortality factor, with the annual period factor defined as the intra-year average of the monthly factors. The latent monthly factor follows a seasonal ARIMA process, and parameters are estimated by maximum likelihood using an EM algorithm with Kalman filtering and smoothing. This setup enables real-time intra-year updates of the latent state and forecasts as new monthly observations arrive without re-estimating model parameters. Using U.S. data for ages 20--90 over 1999--2019, we evaluate intra-year annual nowcasts and one- to five-year-ahead forecasts. The MF--SS model produces both a direct annual forecast and an annual forecast implied by aggregating monthly projections. In our application, the aggregated monthly forecast is typically more accurate. Incorporating monthly information substantially improves intra-year annual nowcasts, especially after the first few months of the year. As a benchmark, we also fit separate annual and monthly Lee--Carter models and combine their forecasts using temporal reconciliation. Reconciliation improves these independent forecasts but adds little to MF--SS forecasts, consistent with MF--SS pooling information across frequencies during estimation. The MF--SS aggregated monthly forecasts generally outperform both unreconciled and temporally reconciled Lee--Carter forecasts and produce more cautious predictive intervals than the reconciled Lee--Carter approach.

Figures (9)

• Figure 1: Annual mortality rates and scaled monthly death rates ($\times 12$) on a log scale, 1999--2019, averaged over ages 20-90.
• Figure 2: Fitted age factor $\hat{a}_{i,x}$ and age-specific loadings $\hat{b}_{i,x}$ for SS and independent LC models, using data from 1999 to 2019 for ages 20-90.
• Figure 3: Fitted period factor $\hat{k}_{i,\tau}$ for SS and independent LC models, using mortality from 1999 to 2019 for ages 20-90.
• Figure 4: Average Kalman gain by age for the annual and monthly specifications, averaged across the fitting period 1999--2019.
• Figure 5: Left: Nowcast MAPE for forecast year 2015. Right: Average nowcast MAPE across expanding windows with forecast years 2015–2019.