Lee-Carter Model¶

Model Specification¶

The Lee-Carter (1992) model:

\[\log(\mu_{xt}) = \alpha_x + \beta_x \kappa_t\]

with identifiability constraints \(\sum_x \beta_x = 1\) and \(\sum_t \kappa_t = 0\).

Fitting¶

import pystmomo as ps

data = ps.load_ew_male()
fit = ps.lc().fit(data.deaths, data.exposures,
                  ages=data.ages, years=data.years)

Inspecting Parameters¶

import numpy as np

# α_x: baseline log-mortality by age
print(fit.ax)          # shape (35,)

# β_x: age sensitivity to the period index
print(fit.bx[:, 0])   # shape (35,)

# κ_t: period (time) index
print(fit.kt[0])       # shape (51,)

# Check constraint: sum(β_x) = 1
print(np.sum(fit.bx))  # ≈ 1.0

Visualising¶

ps.plot_parameters(fit)

This produces a 3-panel chart: \(\alpha_x\) vs age, \(\beta_x\) vs age, \(\kappa_t\) vs year.

Forecasting & Simulation¶

fc   = ps.forecast(fit, h=20)
sim  = ps.simulate(fit, nsim=1000, h=20, seed=42)
boot = ps.semiparametric_bootstrap(fit, nboot=500, seed=0)

References¶

Lee, R.D. & Carter, L.R. (1992). Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association, 87(419), 659–671.