Testing For A Unit Root In Lee-Carter Mortality Model For The Mortality Index

Longevity risk is the risk that actual life spans of individuals or of whole popula-tions will exceed expectations. In the last few decades, life expectancy has considerably increased, and this has obvious benefits. However, there are financial costs of longer life expectancy. Governments that sponsor old-age social security systems, and insurers that sell annuities would have to pay benefits longer than anticipated. Unexpected longevity may also be a financial risk to individuals without guaranteed retirement benefits. In order to manage longevity risk successfully, more advanced and accurate techniques in modeling, securitization and regulation shall be developed. Understanding and fore-casting mortality trend are basic and essential in hedging longevity risk. Since Lee and Carter (1992) described their extrapolative method to model and forecast U.S. death rates, it has become the standard model for the longevity forecast literature and widely adopted for long-term forecasts of age-specific mortality rates. The Lee-Carter mortal-ity model involves a two-step estimation procedure. Empirical findings from using the Lee-Carter model and its extensions prefer an ARIMA(p,1,q) model for modeling the dynamics of the logarithms of mortality rates, which is called mortality index and is a key element in forecasting mortality rates and managing longevity risks.Chapter 1 serves as an introduction. In Chapter 2, we unfortunately proved that the two-step inference procedure in Lee and Carter (1992) may lead to a wrong identi-fication of the dynamics of mortality index if it is not a nearly integrated AR(1) model, which means that future mortality projections based on the two step inference proce-dure for Lee-Carter model and its extensions are questionable. A simulation study and real data analysis are conducted to support the theoretical findings on blindly using the Lee-Carter model. Results in Chapter 2 raise an interesting question on how to test effi-ciently whether the mortality index really follows from a nonstationary AR(1) process, before a sound application of the two-step inference procedure for the Lee-Carter model and its extensions. This will be a challenging future project since the mortality index is unobservable.Motivated by the discovery in Chapter 2 that the two-step inference for Lee-Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for unit root in a Lee-Carter model with an AR(p) process for the mortality index in Chapter 3. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed here are unconventional due to the special structure of errors. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the US mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee-Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is alsp conducted to examine the finite sample behavior of the proposed test.In Chapter 3, the structure of errors in unit root test satisfies the mixing dependence assumption of Phillips & Durlauf (1986). In finance and economics, the GARCH pro-cess is preferred and widely used for modeling error structures, when relating to the unit root issue with non-i.i.d. errors. Chan and Zhang (2010) studied a unit root test based on the least squares estimator for an AR(1) model with GARCH(1,1) errors, where the asymptotic limit depends on the moments of the errors and is nonnormal if the error has an infinite variance. Zhang and Ling (2015) showed that the least squares estimator for a stationary AR model with G-GARCH errors is inconsistent if the error has an infinite variance. Therefore, it is useful to provide a unified unit root test without requiring any prior on the moments of the errors and a consistent estimator for the stationary case. Motivated by the unified empirical likelihood inference in Chan, Li and Peng (2012), Chapter 4 proposes a unified empirical likelihood test for testing unit root in an AR(1) model with GARCH(p,q) errors, whose limit is always a chi-squared distribution with one degree of freedom. Furthermore, an empirical likelihood inference is provided for a stationary AR process with GARCH(p,q) errors, which results in consistent estimation regardless of the heaviness of the tails. The proposed estimator is different from the self-weighted least absolute deviations estimator in Zhu and Ling (2015), where the in-novation in the GARCH errors is assumed to have median zero instead of mean zero. A simulation study confirms the good finite sample performance of the proposed methods before we apply them to some real data sets in finance.