Long-term Prediction of Age-specific Mortality with Multivariate Autoregressive Model
Tamotsu Ooba
Age and sex-specific mortality rates from 1995 to 2030 in Japan were predicted with multivariate autoregressive (MAR) model. This model is of the form,
where 
is a vector of time series, 
is a vector of a variance of order I and -N (O,
), and I is the number of time series those were considered simultaneously, and 
is an AR coefficient matrix of order m .
The MAR model was estimated by the least square method with the Householder reduction with the mortality rates during the period from 1947 to 1 994. The age (O to 84 years) and sex-specific mortality rates were log-transformed and their three types of differential, by age, by year, and by cohort, were calculated. These three differentials and the log-transformed mortality rates were compared and the differential by cohort was selected for the forecast. The differential by year was used supplementary because the differential by cohort does not induce the mortality rate at age O.
Each differential for cohort was converted into the normalized form. Then, was estimated. Long-term prediction of the differentials were made by applying to the differentials repeatedly. The predicted values were reversing transformed into the original form. The death probabilities in the ages 85 years and over were extrapolated toward the direction of the cohort with the nonlinear least square method for the Gompertz-Makeham curve.
The result suggested that the male life expectancy will reach 81.9 years in 2030, and the female life expectancy will reach 88.8 years in 20 16 and thereafter decrease slowly. The estimated growth of the life expectancy is contributed with further reduction of mortality rates in elder ages but not younger ages. On the other hand, the result showed that the declining speed of the female mortality for 20 to 65 years old will be slower, and then, their mortality will increase.
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