Girosi King 2007 - Universal Increasing Nonsmoothness of Age Profiles

Summary

Lee-Carter and RWD forecasts are linear in time, so age-group lines fan out: any quadratic nonsmoothness ${\tilde{\mu }}_t^{\prime }W{\tilde{\mu }}_t={\gamma }_t^2{\beta }^{\prime }W\beta \propto {\gamma }_t^2$ attains a minimum when ${\gamma }_t=0$ and then grows without bound, forecasting forward or backward. This is a property of the model, not the data, and applies for first or second differences, any n-th derivative, or deviation from any fixed baseline age profile.

Main Content

Nonsmoothness measure (Eq. 21)

$$\text{Nonsmoothness}({\mu }_t)=\frac{1}{A-1}\sum _{a=2}^A({\mu }_{at}-{\mu }_{a-1,t}{)}^2.$$

Related to the variance of the profile. Second differences are indifferent to level and slope shifts (“swoosh rotation”). Generalisation: any quadratic form ${\tilde{\mu }}_t^{\prime }W{\tilde{\mu }}_t$ with $W⪰0$; for first differences $W$ is the tridiagonal matrix with ones on the diagonal (ends handled) and $-1$ on the adjacent bands.

Universal increasing nonsmoothness (Eqs. 20-24)

Let ${\mu }_{at}-{\mu }_{a{t}_0}=({\gamma }_t-{\gamma }_{t_0}){\beta }_a$ (Eq. 20) and ${\tilde{\mu }}_t={\gamma }_t\beta$ the mean-adjusted profile. Then

$$\text{Nonsmoothness}({\tilde{\mu }}_t)={\tilde{\mu }}_t^{\prime }W{\tilde{\mu }}_t={\gamma }_t^2{\beta }^{\prime }W\beta \propto {\gamma }_t^2.$$

If ${\gamma }_t$ is monotone in $t$, nonsmoothness decreases until ${\gamma }_t=0$ and thereafter increases for ever; “no feature of the data or forecasting method for projecting the ${\gamma }_t$‘s can make it stop”. Because $\Vert \beta \Vert =1$, Rayleigh’s characterisation gives the lower bound ${\beta }^{\prime }W\beta \ge {\lambda }_1$ (first non-zero eigenvalue of $W$), so $\text{Nonsmoothness}\ge {\lambda }_1{\gamma }_t^2$ independent of the shape of $\beta$.

Proof sketch. Quadratic form is homogeneous of degree 2 in the scalar ${\gamma }_t$; monotone ${\gamma }_t$ crosses zero at most once and $|{\gamma }_t|\to \infty$ along the linear forecast. $\square$

Why. Mixed derivative ${\partial }^2{\mu }_{at}/\partial t\partial a={\gamma }_t^{\prime }{\beta }_a^{\prime }$. With monotone ${\gamma }_t$, the sign of age-slope change is the sign of the derivative of ${\beta }_a$ in age; ${\beta }_a$ is rarely monotone in age (not one instance found), so some neighbouring ages move in opposite directions.

Empirical patterns (Figures 3-4, 6 data sets). Lines fan out: all-cause male New Zealand and Hungary, digestive disease (f) Hungary; lines converge, cross, then fan out: U.S. male suicide (age-profile inversion); near-parallel exception: Portugal male transport accidents. Implausible results: New Zealand 20-year-olds dying more than 40-year-olds; U.K. cervix cancer age profile loses biological plausibility. Figure 5 shows the U-shaped nonsmoothness curve for forecasts (to 2060) and backcasts (to 1900) against in-sample data (U.S. female cardiovascular, Hungary male all-cause, U.S. male all-cause, Norway female cardiovascular, where one component explains only 29%).

Consequences. Cannot be used to study age-profile flattening (Lee and Miller 2001); cannot model convergence to a model life table (subtract baseline and nonsmoothness still grows); exaggerates differences in spending allocation; conflicts with insurers’ graduation practice. Acceptable only with external knowledge that the profile should become less smooth.

Examples

U-shaped nonsmoothness in a synthetic Lee-Carter forecast

Setup: synthetic 17-age, 50-year data as in the estimation note. Compute $\mu =\gamma \hat{\beta }$ at forecast horizons $-\mathrm{100..100}$ in steps of 25 using first-difference $W$. Result: nonsmoothness $0.092,0.042,0.012,0.0001,0.008,0.035,0.081,0.147,0.231$; minimum near the horizon where $\hat{\gamma }=0$ (about 23 years before the last observation). Interpretation: at a horizon of 100 years the profile is orders of magnitude rougher (0.23 against about 0.0001) than at its minimum.

import numpy as np
def nonsmooth(mu):                        # first-difference measure of Eq. 21
    return np.mean(np.diff(mu)**2)
# For tilde-mu = gamma*beta: nonsmooth = gamma**2 * np.mean(np.diff(beta)**2)  -> quadratic in gamma

Connections

Girosi King 2007 - Lee-Carter as a Random Walk with Drift, Girosi King 2007 - Second-Stage Reestimation and Wilmoth Estimators, Forecast Evaluation and Backtesting.

See Also

Girosi King 2007 - Understanding the Lee-Carter Method - Overview