Working with constraints

Regular readers of this blog will be aware of the importance of stochastic mortality models in insurance work. Of these models, the best-known is that from Lee & Carter (1992):

\[ \log \mu_{x,y} = \alpha_x + \beta_x\kappa_y\qquad(1)\]

where \(\mu_{x,y}\) is the force of mortality at age \(x\) in year \(y\) and \(\alpha_x\), \(\beta_x\) and \(\kappa_y\) are parameters to be estimated. Lee & Carter used singular value decomposition (SVD) to estimate their parameters, but the modern approach is to use the method of maximum likelihood — by making an explicit distributional assumption for the number of deaths, the fitting process can make proper allowance for the amount of information available at each age. Once the parameters have been estimated, forecasting takes place by making an assumption over the process driving \(\kappa_y\), e.g. it follows a random walk with drift or an ARIMA process.

Regardless of how the Lee-Carter model is fitted, however, the analyst must also set identifiability constraints. These are necessary because the model specified in Equation (1) has an infinite number of possible solutions. To see this, consider the following simple transformation:

\[ \beta'_x = c\beta_x \\ \kappa'_y = \kappa_y/c\]

for some non-zero constant \(c\) (we previously discussed a very specific instance of this when \(c=-1\)). As is clear from putting \(\beta'_x\) and \(\kappa'_y\) into Equation (1), this will result in an identical fitted value for \(\log \mu_{x,y}\). To tie things down, we need a constraint on the parameter values to counter this infinite number of transformations.

As it happens, there is another such transformation possible involving \(\alpha_x\), so in order to fit the Lee-Carter model we actually need two constraints on the parameters to make their values unique. For example, the constraint system used by Lee & Carter (1992) is as follows:

\[\sum_y\kappa_y = 0\\ \sum_x\beta_x = 1\]

where summation is over the data, i.e. the summation over \(y\) is over the fitted values of \(\kappa_y\) only and does not include the projected values; this permits the fitted parameters to be invariant to the forecast.

The constraint system from Lee & Carter (1992) is the most commonly used one, not least because it is linear. However, there are other constraint systems, such as the one used by Girosi & King (2008):

\[\sum_y\kappa_y = 0\\ \sum_x\beta_x^2 = 1\]

and also the constraint system used by Richards & Currie (2009):

\[\sum_y\kappa_y = 0\\ \sum_y\kappa^2_y = 1\]

These latter two constraint systems are non-linear, but offer some technical advantages in the fitting process. For example, the constraints used by Richards & Currie (2009) were chosen because they fitted the Lee-Carter model as paired Generalized Linear Models (GLMs): conditioning on the values of \(\hat\kappa_y\), \(\hat\alpha_x\) and \(\hat\beta_x\) can be estimated from fitting a GLM, then a second GLM can be used to obtain updated estimates of \(\hat\kappa_y\) by conditioning on the revised values of \(\hat\alpha\) and \(\hat\beta_x\). Repeating this pairwise procedure leads to rapid convergence of the parameter estimates, and with this algorithm it is convenient to apply both the constraints to \(\hat\kappa_y\) in the second step.

The choice of constraint system is up to the analyst fitting the model — the fitted values of \(\log \mu_{x,y}\) will be the same in all three cases, although the parameter estimates will be different. The constraints an analyst chooses will depend on personal preferences, and the above three systems are not the only options. However, it seems to make most sense to stick to those constraint systems that have been published in either textbooks or the peer-reviewed literature.