Lee-Carter Model

Definition

The Lee-Carter model (Lee and Carter 1992) is a parametric stochastic mortality model that decomposes log central death rates into a time-invariant age pattern plus a single time-varying factor multiplied by an age-specific sensitivity:

$\ln(m_{x,t}) = \alpha_x + \beta_x \kappa_t + \varepsilon_{x,t}$

where $m_{x,t}$ is the central death rate at age $x$ and time $t$ , $\alpha_x$ captures the average age-specific mortality profile, $\beta_x$ captures how sensitive each age is to overall mortality improvements, and $\kappa_t$ is the unobserved time index of mortality (the "mortality level"). The error $\varepsilon_{x,t}$ is assumed white noise.

Key Ideas

The model reduces a high-dimensional matrix of death rates to a single latent time series $\kappa_t$ , making forecasting tractable.
$\kappa_t$ is modeled as a random walk with drift: $\kappa_t = \kappa_{t-1} + \mu + \epsilon_t$ , so mortality improvements continue at rate $\mu$ indefinitely.
Parameter identification requires two constraints; Lee and Carter use $\sum_x \beta_x = 1$ and $\sum_t \kappa_t = 0$ .
The model is fit to historical data by singular value decomposition (original Lee-Carter 1992) or maximum likelihood under Poisson counts.

How It Works

Single-Population Model

Given historical age-period death rates and exposures $\{m_{x,t}, E_{x,t}\}$ , estimate $\alpha_x$ , $\beta_x$ , and $\kappa_t$ jointly. Under Poisson likelihood, the death count $D_{x,t} \sim \mathrm{Poisson}(m_{x,t} E_{x,t})$ , and the log-likelihood is:

$\ell = \sum_{x,t} \left[ D_{x,t}(\alpha_x + \beta_x\kappa_t + \ln E_{x,t}) - E_{x,t} e^{\alpha_x + \beta_x\kappa_t} \right]$

The fitted $\hat\kappa_t$ series is then modeled as a univariate time series (typically a random walk with drift) to produce stochastic mortality forecasts and confidence intervals.

Two-Population Extension (Zhou et al.)

For two related populations sharing the constraint $\beta_x^{(1)} = \beta_x^{(2)}$ (identical age sensitivities), the model becomes:

$\ln(m_{x,t}^{(i)}) = \alpha_x^{(i)} + \beta_x \kappa_t^{(i)}, \qquad i = 1, 2$

The shared $\beta_x$ embeds the non-divergence hypothesis: if mortality improvements are driven by the same age-pattern forces, the long-run forecasts of $m_{x,t}^{(1)}$ and $m_{x,t}^{(2)}$ cannot diverge in direction. The second step models $(\kappa_t^{(1)}, \kappa_t^{(2)})$ jointly with a bivariate time-series process.

Three options for the joint $\kappa_t$ dynamics:

RWAR (random walk + autoregressive) (Cairns et al. 2011): $\kappa_t^{(1)}$ = random walk; spread $\kappa_t^{(1)} - \kappa_t^{(2)}$ = autoregressive order 1 (AR(1)). Requires assuming population 1 is dominant.
VAR (vector autoregression): symmetric VAR(1) in differences $\Delta\kappa_t^{(i)}$ with an explicit non-divergence constraint.
VECM (vector error correction model): symmetric VECM with error-correction terms $\rho^{(i)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)})$ . Non-divergence is automatic: the long-run equilibrium $\kappa_t^{(1)} - \kappa_t^{(2)} \to$ constant is enforced by the error-correction mechanism. See Cointegration.

Fertility Extension (Lee 1993)

Lee (1993) extends the same factor-model logic to fertility. Age-specific fertility rates are modeled as:

$f_{i,t} = \alpha_i + \beta_i F_t + u_{i,t}$

where $\alpha_i$ is the mean age- $i$ fertility rate, $\beta_i$ the age- $i$ sensitivity, and $F_t$ the total fertility factor (analogous to $\kappa_t$ for mortality). The total fertility rate satisfies $\mathrm{TFR}_t = F_t + A$ for a constant $A$ .

The dynamics of $F_t$ are specified as an autoregressive moving average (ARMA)(1,1) constrained to a steady state of 2 (replacement level):

$F_t = 2(1-\phi) + \phi F_{t-1} + e_t + \theta e_{t-1}$

The long-run mean $E[F_t] = 2$ encodes the demographic belief that fertility converges to replacement level; $\phi$ and $\theta$ govern persistence and smoothing of the path. In the Meseguer (2010) Bayesian VAR (BVAR) implementation, this prior is encoded through a tight prior autoregressive (AR) mean of 0.8 for fertility (rather than 0 for mortality), combined with a tight overall shrinkage $\lambda_1 = 0.05$ .

Estimation in Two Steps

Because Lee-Carter is nonlinear in $(\alpha_x, \beta_x, \kappa_t)$ , the full model is estimated in two steps:

Fit the Lee-Carter parameters $(\alpha_x, \beta_x, \kappa_t^{(i)})$ jointly via maximum likelihood estimation (MLE).
Fit the bivariate time-series process to the estimated $\hat\kappa_t^{(i)}$ series via MLE.

Two-step estimation introduces parameter uncertainty in step 2 that must be corrected using Murphy-Topel (2002) standard errors.

Limitations and BVAR Comparison (Meseguer 2010)

The Lee-Carter framework has several structural limitations highlighted by comparison with BVAR approaches:

Single uncertainty source. The random walk in $\kappa_t$ is the only stochastic component — parameters $\alpha_x$ and $\beta_x$ are treated as fixed constants. This understates total forecast uncertainty, particularly at long horizons where parameter uncertainty compounds.
Fixed age sensitivity ratios. The relative mortality improvement rates across ages are $\beta_x/\beta_j$ — a constant ratio fixed at estimation. BVAR allows each age group its own shock and correlation structure, reflecting that age groups can diverge over long horizons.
Calibration failure at long horizons (Meseguer 2010). With a 1995 base year, the actual 65+ female population is already outside Lee-Carter's 90% credible interval by 2009 — a 14-year horizon. The informative BVAR maintains better-calibrated intervals throughout.
No information sharing across age groups in forecasting. Lee-Carter's second-step univariate AR for $\kappa_t$ uses no information about the cross-age-group correlation structure. The BVAR Minnesota prior with correlation-adaptive $\lambda_2$ explicitly borrows strength across adjacent age groups.
Fertility. Lee-Carter is a mortality model only; the Lee (1993) fertility extension applies the same single-factor structure ( $F_t$ instead of $\kappa_t$ ) without the non-divergence guarantees available from VECM-type models.

Why It Matters

The Lee-Carter model is the dominant paradigm in stochastic mortality forecasting and the foundation of most actuarial mortality projection work for pension and annuity liabilities.
The reduction to a single time series $\kappa_t$ enables standard time-series tools (autoregressive integrated moving average (ARIMA), VAR, VECM) to be applied directly to mortality forecasting.
The two-population extension with VECM provides a principled, symmetric framework for modeling longevity basis risk, avoiding the subjective dominant-population assumption.
Mortality forecasts from the model feed directly into Solvency II capital calculations (Solvency Capital Requirement (SCR) for longevity risk; risk margin).

Open Questions

The single time factor $\kappa_t$ may be insufficient for populations with strong cohort effects (e.g., UK data shows the 1920s cohort with unusually rapid mortality improvement).
The random walk assumption for $\kappa_t$ implies ever-increasing uncertainty in long-horizon forecasts; bounded mortality models (logistic transforms) are alternatives.
When $\beta_x^{(1)} \neq \beta_x^{(2)}$ , non-divergence requires more complex restrictions and VECM alone is insufficient.