Lee-Carter Model Parameter Estimation with SVD

The Estimation Challenge

Let's recall the model equation:

The terms on the right-hand side, $a_x$, $b_x$, and $k_t$, are not directly observable. We cannot simply use ordinary least squares regression because the term $b_x{k}_t$ involves a product of two unknown parameters. The solution, pioneered by Lee and Carter, is a multi-stage approach where we first estimate $a_x$, and then use SVD to solve for $b_x$ and $k_t$ simultaneously.

A Refresher on Singular Value Decomposition (SVD)

Before diving into the estimation steps, it's worth briefly recapping the essence of SVD. At its heart, SVD decomposes any matrix $Z$ into three other matrices: $Z=U\Sigma V^T$.

• $U$ and $V$ are orthogonal matrices whose columns are called singular vectors.
• $\Sigma$ is a diagonal matrix containing the singular values, which are positive and ordered from largest to smallest.

The power of SVD lies in its ability to find the best low-rank approximation of a matrix. The Lee-Carter model postulates that the variation in mortality improvements over time can be captured by a single factor ($k_t$). This is equivalent to saying that the matrix of mortality improvements, after accounting for the average age profile, is approximately a rank-1 matrix. SVD is the perfect tool for finding this best rank-1 approximation.

For a deeper mathematical overview of SVD, the following video provides an excellent and concise explanation.

The SVD Estimation Procedure

We can now outline the estimation procedure as a clear, step-by-step process. This method directly applies the SVD concept you just reviewed. A detailed guide is provided in the following document, which we will walk through.

Let's synthesize those steps into a coherent procedure.

Step 1: Estimate $a_x$

The parameter $a_x$ represents the average mortality profile for each age. If we assume the errors ${ϵ}_{x,t}$ average to zero over time and we impose the constraint ${\sum }_t{k}_t=0$, taking the average of the model equation over time $t$ gives us a simple estimator for $a_x$:

Here, $T$ is the total number of years in our data. ${\hat{a}}_x$ is simply the arithmetic mean of the log death rates for age $x$ across all years.

Step 2: Form the Residual Matrix

Once we have ${\hat{a}}_x$, we can subtract it from our log-mortality data. This leaves us with a matrix that should be explained by the $b_x{k}_t$ term plus noise. Let's call this matrix $Z$:

The matrix $Z$ has ages ($x$) for rows and time ($t$) for columns.

Step 3: Apply SVD to the Matrix $Z$

This is the core of the estimation. We perform SVD on the matrix $Z$:

Here, $U_i$ and $V_i$ are the column vectors of $U$ and $V$, and ${\sigma }_i$ are the singular values. The Lee-Carter model assumes a rank-1 structure for the mortality improvements, so we use the first term of the SVD as our estimate:

where $U_{x,1}$ is the $x$-th element of the first left singular vector, and $V_{t,1}$ is the $t$-th element of the first right singular vector.

Step 4: Extract and Scale $b_x$ and $k_t$

By comparing $b_x{k}_t$ with ${\sigma }_1{U}_{x,1}{V}_{t,1}$, we can identify the parameters. A common approach is to set:

• ${\hat{b}}_x=U_{x,1}$ (the first left singular vector)
• ${\hat{k}}_t={\sigma }_1{V}_{t,1}$ (the first right singular vector, scaled by the first singular value)

This choice naturally satisfies the constraint ${\sum }_x{\hat{b}}_x^2=1$, since the columns of $U$ are orthonormal. This is a slightly different constraint from the ${\sum }_x{b}_x=1$ we discussed previously, but it is a common convention when using SVD.

There's a small but important practical detail: the signs of the singular vectors $U_1$ and $V_1$ are arbitrary. SVD could return $-U_1$ and $-V_1$, and the product would be the same. We need the $k_t$ series to show a downward trend for mortality improvement. If the estimated ${\hat{k}}_t$ goes up over time, you simply flip the signs of both ${\hat{b}}_x$ and ${\hat{k}}_t$.

You can find this procedure summarized clearly in this MathWorks documentation.

Step 5: Enforce the Final Constraint

We still need to enforce the constraint ${\sum }_t{k}_t=0$. The ${\hat{k}}_t$ we just estimated will not generally sum to zero. We can enforce this by centering the ${\hat{k}}_t$ series and making a corresponding adjustment to ${\hat{a}}_x$.

1. Calculate the mean of our estimated mortality index: $\bar{k}=\frac{1}{T}{\sum }_t{\hat{k}}_t$.
2. Create the final, centered mortality index: $k_t^{final}={\hat{k}}_t-\bar{k}$.
3. Adjust the average age profile to compensate: $a_x^{final}={\hat{a}}_x+\bar{k}{\hat{b}}_x$.

After these adjustments, we have our final set of estimated parameters that satisfy the necessary constraints for identifiability.

The Result: An Estimated Time Series

The output of this entire process is a set of estimated vectors ${\hat{a}}_x$ and ${\hat{b}}_x$, and most importantly, a historical time series for the mortality index, ${\hat{k}}_t$. As Ronald Lee mentions in his talk, a remarkable and surprising finding is how often this estimated ${\hat{k}}_t$ series turns out to be close to a straight line.

The plot he shows is exactly what our estimation procedure produces. It is this time series that we will carry forward to the next stage of our analysis.

Conclusion

In this lesson, we have bridged the gap between the theoretical structure of the Lee-Carter model and its practical application. We've established a robust, step-by-step procedure for fitting the model to historical data.

Key Takeaways:

• Estimation Strategy: The Lee-Carter parameters are estimated in stages, not with a single regression.
• Role of SVD: Singular Value Decomposition is the key tool used to separate the age-specific sensitivities ($b_x$) from the time-varying mortality index ($k_t$).
• Rank-1 Approximation: The method works because the Lee-Carter model is fundamentally a rank-1 approximation of the matrix of mortality improvements.
• Estimation Steps:
  1. Estimate ${\hat{a}}_x$ by averaging log-death rates over time.
  2. Form the residual matrix $Z=\ln (m_{x,t})-{\hat{a}}_x$.
  3. Apply SVD to $Z$ to get $U,\Sigma ,V$.
  4. Extract initial estimates: ${\hat{b}}_x\leftarrow U_1$ and ${\hat{k}}_t\leftarrow {\sigma }_1{V}_1$.
  5. Adjust ${\hat{a}}_x$ and ${\hat{k}}_t$ to enforce the constraint ${\sum }_t{k}_t=0$.

Preview of the Next Lesson

We have successfully extracted the historical trend of mortality improvement, embodied in the time series ${\hat{k}}_t$. The final step in creating a full projection model is to forecast this trend into the future. In our next lesson, we will leverage your strong background in time series analysis to model and project the ${\hat{k}}_t$ series using ARIMA models, completing the Lee-Carter forecasting methodology.