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CPS2210 Justyna Majewska et al.
general – is the best in order to project mortality rates. The best model for one
country does not mean that this model will be the best for the other.
Most work has focused on stochastic mortality models for single
populations, however, it is important to be able to model two or more
populations simultaneously. As Li and Lee (2005) indicated the convergence
in mortality levels for closely related populations can lead to unsuitable
mortality projections, if the projections for individual populations are obtained
in isolation from one another. Similar historical trends in long-run life
expectancy patterns can be useful for countries. Besides, analyses of the main
determinants of life expectancy (the socio-economic, environmental or
behavioural factors) of associated populations are crucial. Knowledge of
existence of some common stochastic trends in mortality rate in cluster of
European countries can be used for projections mortality rates and life
expectancy (Majewska 2017; Lazar et al. 2016).

2. Methodology
Many models have been proposed in the literature to represent the
mortality evolution of two or more related populations. The majority of such
models extend known single population models by specifying the correlation
and interaction between the involved populations (Villegas et al., 2017). The
Lee-Carter model (1992) was originally developed for a single country, and is
defined as follows:
logm(x,t) =a(x)+b(x)k(t)+e(x,t)
The country specific α(x) determines the baseline shape of the mortality
curve in a country, β(x) (age-specific component) tells us which rates decline
rapidly and which rates decline slowly in response to changes in κ(t) (time-
varying mortality index). ε(x,t) is the error term of Lee-Carter model with mean
zero and variance σ . Mortality index κ(t) is used to forecast the series. Since

δ
parameters β(x) and κ(t) are unobserved variables, the least square estimates

can be found by using the Singular Value Decomposition method.
We adopt an extension of the Lee-Carter method suggested by Li and Lee
(2005), the so-called augumented common factor model. Model is defined as
follows:
logm(x,t,i) =a(x,i)+b(x)k(t,i)+e(x,t,i)
Li and Lee (2005) proposed that the parameters should be estimated using
two-step singular value decomposition, firstly estimating the common
parameters from the combined data for all countries, and secondly estimating
the rest of the country specific parameters. Model takes into account the fact
that mortality patterns for closely related populations are expected to be
similar.
where C1 and C2 donotes populations from two different countries. This

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