Quantifying Mortality Uncertainty with Confidence Intervals

The Anatomy of Uncertainty in Mortality Forecasts

When we project mortality, our forecasts are uncertain for several reasons. A robust analysis requires us to acknowledge and, where possible, quantify each source of error. The image below shows a typical Lee-Carter forecast for life expectancy. The central line is the point forecast we learned to calculate previously, while the shaded blue area represents the prediction interval—our quantification of uncertainty.

So, what contributes to the width of this shaded region? We can break down the total uncertainty into three main components.

As you've just read, the three primary sources are:

1. Process Variance (Poisson Variability): Death is a random event. Even if we knew the "true" underlying mortality rate for a group of people, the actual number of deaths observed would vary from year to year. This is often modelled as a Poisson process, and it represents the inherent randomness of mortality that no model can eliminate.
2. Parameter Estimation Uncertainty: The Lee-Carter parameters $a_x$, $b_x$, and the historical $k_t$ values are not "true" values; they are estimates derived from historical data. A different set of historical data would have produced slightly different parameter estimates. This uncertainty in our estimated parameters contributes to the overall uncertainty of our forecast.
3. Extrapolation Uncertainty: In our previous lessons, you used your expertise in ARIMA modeling to forecast the $k_t$ index. You'll recall that any time series forecast comes with a prediction interval, which widens as the forecast horizon increases. This uncertainty in the future path of $k_t$ is typically the largest contributor to long-term forecast uncertainty.

Most early and simplistic approaches to quantifying uncertainty focused only on the third source. Our goal is to explore more comprehensive methods.

An Analytical Approximation

The simplest way to create a prediction interval for a projected mortality rate, ${\hat{m}}_{x,T+h}$, is to use an approximate formula proposed by Lee and Carter themselves. This method leverages the standard error of the $k_t$ forecast, which is a standard output from an ARIMA model.

The formula for an approximate $100(1-\alpha )$% prediction interval for the log-mortality rate is:

Exponentiating this gives the interval for the mortality rate itself.

While simple to compute, this analytical interval only accounts for the extrapolation uncertainty of the $k_t$ index. It completely ignores the uncertainty from parameter estimation and Poisson process variance. As the Keilman and Pham paper pointed out, this can lead to prediction intervals that are too narrow, understating the true level of risk.

A More Robust Approach: Bootstrapping

To simultaneously account for multiple sources of uncertainty, we turn to simulation methods. Bootstrapping is a powerful computational technique that allows us to approximate the sampling distribution of an estimator or a forecast when its theoretical distribution is intractably complex. Given your statistics background, you're likely familiar with the general principle of resampling from an initial sample to understand variability.

In the context of the Lee-Carter model, we can design a bootstrap algorithm to capture the combined effects of parameter uncertainty and forecast uncertainty.

The Bootstrapping Algorithm for Lee-Carter Forecasts

The core idea is to simulate many possible future mortality paths, and then use the distribution of these paths to construct our prediction intervals. The paper by Keilman and Pham provides a clear algorithm, which we can summarize in a few key steps.

Let's distill the process you just read about:

1. Fit the Initial Model: Fit the Lee-Carter model to the historical data to get initial estimates ${\hat{a}}_x,{\hat{b}}_x,{\hat{k}}_t$ and the matrix of residuals, ${\widehat{ϵ}}_{x,t}$.
2. Bootstrap the Residuals (or Data): Generate a large number of bootstrap samples. There are a few ways to do this:
   • Residual Bootstrap: Create new matrices of log-mortality rates by adding resampled residuals (with replacement) to the fitted values: $\ln (m_{x,t}^{\ast })={\hat{a}}_x+{\hat{b}}_x{\hat{k}}_t+{ϵ}_{x,t}^{\ast }$.
   • Poisson Bootstrap (as in the paper): Simulate a new matrix of death counts $D_{x,t}^{\ast }$ from a Poisson distribution where the mean is based on the fitted rates from the initial model. This is generally preferred as it more directly models the data generating process.
3. Re-estimate and Forecast: For each of the bootstrap samples of death data (let's say we have 1,000 of them):
   a. Re-estimate: Fit the Lee-Carter model to the bootstrapped data $D_{x,t}^{\ast }$ to get a new set of parameters: ${\hat{a}}_x^{\ast },{\hat{b}}_x^{\ast },{\hat{k}}_t^{\ast }$. The variation across these 1,000 sets of parameters reflects the parameter uncertainty.
   b. Re-forecast: Fit an ARIMA model to the new historical index ${\hat{k}}_t^{\ast }$. Use this model to simulate one future path for the index, ${\hat{k}}_{T+h}^{\ast }$. This step incorporates the forecast uncertainty.
4. Generate Forecast Distribution: Using each set of bootstrapped parameters and its corresponding forecasted index, calculate a full schedule of future mortality rates: ${\hat{m}}_{x,T+h}^{\ast }=\exp ({\hat{a}}_x^{\ast }+{\hat{b}}_x^{\ast }{\hat{k}}_{T+h}^{\ast })$. Also calculate the corresponding life expectancy, ${\hat{e}}_x^{\ast }$.
5. Construct Prediction Intervals: You now have 1,000 simulated values for each future mortality rate and life expectancy. The 95% prediction interval is simply the range between the 2.5th percentile and the 97.5th percentile of this simulated distribution.

This simulation-based approach provides a far more complete picture of the total uncertainty than the simple analytical formula.

Decomposing and Understanding the Uncertainty

A major advantage of this simulation approach is that we can isolate the different sources of uncertainty to see how much each one contributes to the total.

The key findings from this analysis are crucial for any practitioner:

• Long-Term vs. Short-Term: In the short run (the first few years of the forecast), uncertainty from the model fit (parameter uncertainty and process variance) can be a substantial part of the total forecast variance.
• Dominance of Trend Uncertainty: In the long run, the uncertainty from extrapolating the time index $k_t$ becomes the dominant source of error. The "fan chart" of the forecast widens primarily due to the uncertainty in the future trend of mortality improvement.
• Ignoring Sources Matters: The paper demonstrates that ignoring Poisson variability (by using Ordinary Least Squares instead of a method like Weighted Least Squares or a Poisson GLM) can lead to biased forecasts and incorrect intervals. Similarly, ignoring parameter uncertainty leads to intervals that are significantly too narrow.

For an actuary advising a pension fund that needs to provision for liabilities 50 years into the future, the clear message is that the uncertainty around the long-term trend of mortality improvement ($k_t$) is the single biggest risk factor.

Conclusion

In this lesson, we have moved from the deterministic world of point forecasts into the stochastic reality of risk management. We have seen that quantifying the uncertainty in mortality projections is as important as calculating the central estimate itself.

Key Takeaways:

• Sources of Uncertainty: Lee-Carter forecasts are subject to three main sources of error: inherent randomness in deaths (process variance), uncertainty in model parameter estimates, and uncertainty in the time series forecast of the mortality index.
• Analytical vs. Simulation: A simple analytical formula can provide a quick approximation of the prediction interval, but it only accounts for the $k_t$ forecast error and understates the true uncertainty.
• Bootstrapping: A simulation-based bootstrap approach is the state-of-the-art method for constructing comprehensive prediction intervals. It can be designed to capture all major sources of uncertainty simultaneously.
• Dominance of Trend Uncertainty: For long-term projections, uncertainty in the future trajectory of the $k_t$ index is the most significant driver of the overall forecast uncertainty.

Preview of the Next Lesson

The Lee-Carter model, while foundational, is just one of many tools available for mortality projection. It has known limitations, such as the assumption that the pattern of age-specific mortality improvement is constant over time. In our next and final lesson of this module, we will broaden our perspective by describing and comparing alternative mortality projection models, such as the Age-Period-Cohort (APC) family of models and P-spline methods, which aim to provide more flexible and potentially more accurate forecasts.