A Flexible Functional Form Approach To Mortality Modeling: Do We Need Additional Cohort Dummies?

• Author: Colin O'hare,Han Li,Farshid Vahid
• Published date: 01 July 2017
• DOI: http://doi.org/10.1002/for.2437
• Date: 01 July 2017

Journal of Forecasting,J. Forecast. 36, 357–367 (2017)

Published online 16 August 2016 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2437

A Flexible Functional Form Approach To Mortality Modeling: Do

We Need Additional Cohort Dummies?

HAN LI,COLIN O’HARE AND FARSHID VAHID

ABSTRACT

The increasing amount of attention paid to longevity risk and funding for old age has created the need for precise

mortality models and accurate future mortality forecasts. Orthogonal polynomials have been widely used in technical

fields and there have also been applications in mortality modeling. In this paper we adopt a flexible functional form

approach using two-dimensional Legendre orthogonal polynomials to fit and forecast mortality rates. Unlike some

of the existing mortality models in the literature, the model we propose does not impose any restrictions on the age,

time or cohort structure of the data and thus allows for different model designs for different countries’ mortality

experience. Weconduct an empirical study using male mortality data from a range of developed countries and explore

the possibility of using age–time effects to capture cohort effects in the underlying mortality data. It is found that, for

some countries, cohort dummies still need to be incorporated into the model. Moreover, when comparing the proposed

model with well-known mortality models in the literature, we find that our model provides comparable fitting but

with a much smaller number of parameters. Based on 5-year-ahead mortality forecasts, it can be concluded that the

proposed model improves the overall accuracy of the future mortality projection. Copyright © 2016 John Wiley &

Sons, Ltd.

KEY WORDS mortality; orthogonal polynomials; cohort effects; forecasting

INTRODUCTION

Owing to the rapid growth in life expectancy during the past few decades, longevity risk has now become one of the

most significant risks faced by governments, insurance companies and superannuation funds. Moreover, the increased

focus on life as a risk that can be commoditized and traded through mortality-linked financial markets has also created

the desire to understand and forecast mortality rates more accurately. Therefore, increasing attention has been given

to mortality modeling in recent years.

According to Booth and Tickle (2008), recent attempts at mortality modeling normally treat age and time effects

as the most significant factors that would affect mortality experience (see, for example, Lee and Carter, 1992;

Renshaw et al., 1996; Cairns et al., 2006; Plat, 2009). More recently, an additional perceived pattern in mortality

data, the ‘cohort effect’ (Renshaw and Haberman, 2006) has been brought to our attention. The cohort effect has been

identified in several countries. For example, according to the Government Actuary’s Department (2002) in the UK,

the generations born between 1925 and 1945 have experienced a more rapid decline in mortality rates compared to

other generations. Adding cohort dummies into mortality models has been shown to improve fitting (Renshaw and

Haberman, 2006; Cairns et al., 2009) in some but not all countries. The presence of cohort effects is therefore debat-

able and its justification is limited. Cairns et al. (2011) discussed the possibility that cohort effects could be fully or

partially captured by more complex structure of age and time effects. This has created the primary motivation of our

analysis, which is to gain a better understanding of the cohort effects. In this paper, we propose a two-dimensional (2-

D) Legendre orthogonal polynomials (LOP) model to investigate the feasibility of using age–time effects to capture

cohort effects.

Furthermore, there have been many studies on the comparison of different mortality models and it is concluded

that no single model dominates for all countries (Cairns et al., 2011; Dowd et al., 2010). The result is not surprising

since different countries will have different characteristics in mortality experience, and thus it is less likely for one

model to provide a best fit for all countries. One of the strengths of our flexible functional form approach is the

fact that the model we propose is data driven without prior belief on age, time or cohort structure of the underlying

data. Cross-validation will be used to select age–time effects based on the features of a specific country’s mortality

experience.

Fitting male mortality data for age 50–89 from a range of developed countries over the period 1950–2009, we

conclude that our model achieves better or at least comparable fit quality with a much smaller number of parameters

when compared to the well-known Lee–Carter model and Plat model in the literature. A check of the residual plots at

this stage is done to identify whether cohort additions are necessary and, if so, for which generations and countries.

Correspondence to: Han Li, Department of Econometrics and Business Statistics, Monash University Melbourne, VIC, Australia. E-mail:

han.li@monash.edu

Copyright © 2016 John Wiley & Sons, Ltd