Forecasting the Lee-Carter Time Index with ARIMA

Introduction

In our previous lesson, we successfully navigated the process of estimating the Lee-Carter model parameters from historical data. Using Singular Value Decomposition, we extracted the average age profile of mortality, $a^_{x}$ , the age-specific response to mortality changes, $b^_{x}$ , and, most importantly, the historical time series of the general mortality index, $k^_{t}$ .

Now, we will address the central task of projection. This lesson focuses on taking that historical $k^_{t}$ series and forecasting its future path. Your strong background in time series analysis, particularly with ARIMA models, will be invaluable here. We will not be learning ARIMA from scratch, but rather applying this powerful toolkit to the specific context of the Lee-Carter mortality index. We will explore the common statistical properties of the $k^_{t}$ series and walk through the systematic process of building and using a forecasting model.

The Time Series of Mortality Improvement

After fitting the Lee-Carter model to data from numerous countries and time periods, a remarkable and consistent pattern emerged. The estimated mortality index, $k^_{t}$ , almost invariably shows a strong, persistent linear trend over time.

To understand how to model this, let's turn to a presentation by Ronald Lee himself.

Ronald Lee: The Lee-Carter Model - An Update and Some Extensions

This video features one of the model's creators, Ronald Lee, discussing its properties.

Watch the segment from this discussion of the model's components. Pay close attention to his description of the fitted k_t series. He notes the surprising linearity of the trend and mentions that when analyzed as a time series, it was found to behave as a random walk with drift. This empirical finding forms the foundation of the standard forecasting approach.

The model that captures this behavior is a specific type of ARIMA model, namely an ARIMA(0,1,0) model with a constant. It is defined by the following equation:

k_{t} = k_{t - 1} + d + e_{t}

Let's break this down:

$k_{t}$ is the value of the mortality index at time $t$ .
$d$ is the drift parameter. This constant term represents the average annual change in $k_{t}$ . Since a decrease in $k_{t}$ corresponds to an improvement in mortality, $d$ is typically negative.
$e_{t}$ is a white noise error term, usually assumed to be normally distributed with mean 0 and constant variance $σ_{e}^{2}$ .

This simple structure effectively models a process that has a stochastic linear trend. The following resource provides a concise mathematical summary of this time series model.

[PDF] The Lee-Carter Model - Statistics and Population

This document provides a clear summary of the Lee-Carter methodology.

Please read the section titled The Time Series Model. It formally presents the random walk with drift equation for k_t and discusses the interpretation of the drift d and the error variance v (which we denote as \sigma_e^2).

Building the Forecast Model: The Box-Jenkins Approach

While the random walk with drift is the most common model for $k_{t}$ , it is not a foregone conclusion. As a statistician, you know that a rigorous analysis is required to select the most appropriate model for a given time series. We achieve this by applying the Box-Jenkins methodology, a process you are already familiar with.

The following resource from MathWorks provides an excellent, applied walkthrough of using this methodology to find the best ARIMA model for a $k_{t}$ series. We will use it as a guide.

Forecast Mortality Trends Using Lee-Carter Model - MathWorks

This document details the practical steps for fitting an ARIMA model to the Lee-Carter mortality index.

Please read the sections under the heading Forecast Using ARIMA Models. We will step through this process. Start with the Motivation subsection to understand the rationale for using ARIMA models. Next, study the Preliminary Analysis of k_t. This section demonstrates how to determine the order of integration, d, by using plots and formal tests (ADF and KPSS) to check for stationarity. Notice how they conclude that differencing is necessary. Then, read about the Identification of the Orders (p,q). This shows how to use the ACF and PACF plots of the stationary (differenced) series to select the AR and MA orders. Finally, review Parameter Estimation to see how the model coefficients (like the drift term) are estimated once the model structure is identified.

Let's summarize the procedure you just read about:

Identification:
- Plot $k_{t}$ : Visually inspect the series for trends and other patterns. You will almost certainly see a non-stationary linear trend.
- Differencing and Stationarity Tests: Take the first difference of the series, $Δ k_{t} = k_{t} - k_{t - 1}$ . Use formal tests like the Augmented Dickey-Fuller (ADF) test to confirm if the differenced series is stationary. The null hypothesis of the ADF test is that a unit root is present (the series is non-stationary). We aim to reject this null for the differenced series. If the random walk with drift model is appropriate, the first-differenced series $Δ k_{t}$ will be stationary.
- Select AR and MA orders (p, q): Examine the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) of the stationary (differenced) series. For a pure random walk with drift, the ACF and PACF of $Δ k_{t}$ should show no significant spikes after lag 0.
Estimation:
- Once you have identified the model, for example, an ARIMA(0,1,0) with a constant, you estimate its parameters.
- The constant term in the model for the differenced series, $Δ k_{t}$ , becomes our estimate for the drift, $d^$ .
- The variance of the residuals from this model gives us our estimate for the error variance, $σ^_{e}^{2}$ .
Diagnostic Checking:
- Finally, analyze the residuals of the fitted ARIMA model. They should resemble white noise, meaning they have no discernible autocorrelation. This confirms that your model has adequately captured the systematic patterns in the data.

Projecting $k_{t}$ and Quantifying Uncertainty

With a fitted and validated ARIMA model, we can now project the mortality index into the future.

The point forecast is a straight line extending from the last observed value, with a slope equal to the estimated drift, $d^$ . For a forecast $h$ years into the future from the last data point at time $T$ , the forecast is:

k^_{T + h} = k_{T} + h \cdot d^

However, a point forecast alone is insufficient. Actuarial work requires quantifying the uncertainty around this forecast. This is where prediction intervals come in. A key feature of the random walk model is that the variance of the forecast error increases linearly with the forecast horizon.

The variance of the forecast $k^_{T + h}$ is given by:

Var (k^_{T + h}) = h \cdot σ^_{e}^{2}

This means a 95% prediction interval at horizon $h$ can be calculated as:

k^_{T + h} \pm 1.96 h \cdot σ^_{e}^{2}

Notice how the $h$ term causes the interval to widen over time, reflecting our increasing uncertainty about the distant future. This widening interval creates a "fan chart" effect, which is a hallmark of stochastic mortality projections.

The left panel shows a forecast of life expectancy at birth, which is derived from projected mortality rates. The solid line represents the central forecast, analogous to our projected $\hat{k}_t$. The shaded blue area is the prediction interval, which widens over time, illustrating the increasing uncertainty as we forecast further into the future. The right panel shows the model's fit to death rates for a single year.

The following video provides a practical demonstration of forecasting and simulating future mortality paths, including the generation of these fan charts.

StMoMo: An R Package for Stochastic Mortality Modelling

This presentation on the StMoMo R package demonstrates the forecasting stage of a mortality model.

Watch the segment from the beginning of the forecasting section. The presenter explains that the time-dependent components (our k_t) are modeled using time series methods (a random walk with drift). Observe how this leads to projected mortality rates with prediction intervals, creating the characteristic "fan" of possible future outcomes.

Conclusion

In this lesson, we have connected the historical mortality index, $k^_{t}$ , to its future projection using the robust framework of ARIMA modeling. By applying your existing expertise in time series analysis, we've seen how to formalize the forecasting process for the Lee-Carter model.

Key Takeaways:

Model for $k_{t}$ : The Lee-Carter mortality index, $k^_{t}$ , is empirically found to be well-described by a random walk with drift, which is an ARIMA(0,1,0) model with a constant.
Box-Jenkins Application: The standard Box-Jenkins methodology (Identification, Estimation, Diagnostics) should be used to confirm the model structure for a given dataset. This involves checking for stationarity via differencing and analyzing ACF/PACF plots.
Forecasting: The point forecast for $k_{t}$ follows a linear path determined by the estimated drift parameter, $d^$ .
Uncertainty: Prediction intervals for the forecast widen with the forecast horizon $h$ , as the variance of the forecast error is proportional to $h$ . This is a critical feature for quantifying risk.

Preview of the Next Lesson

We have now estimated all the components of the Lee-Carter model ( $a^_{x}$ , $b^_{x}$ ) and created a stochastic forecast for the time component ( $k^_{t}$ ). In our next lesson, we will put all these pieces together. We will learn how to use our forecast of $k_{t}$ to calculate projected age-specific mortality rates ( $m^_{x, t}$ ) and then use these rates to compute one of the most important outputs of any mortality model: future life expectancy.

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