Trends in the Quantiles of the Life Table Survivorship Function

Abstract

We offer a new approach for modeling past trends in the quantiles of the life table survivorship function. Trends in the quantiles are estimated, and the extent to which the observed patterns fit the unit root hypothesis or, alternatively, an innovative outlier model, are conducted. Then a factor model is applied to the detrended data, and it is used to construct quantile cycles. We enrich the ongoing discussion about human longevity extension by calculating specific improvements in the distribution of the survivorship function, across its full range, and not only at the central-age ranges. To illustrate our proposal, we use data for the UK from 1922 to 2013. We find that there is no sign in the data of any reduction in the pace of longevity extension during the last decades.

Appendix

1.1 Factor Models

In this context, the factor model on the quantile cycles can be presented as:

$$\begin{array}{*{20}c} {c_{t}^{\alpha } = \lambda^{\alpha } \varvec{F}_{t} + {\text{error}}_{t}^{\alpha } } \\ {c_{t}^{\alpha } = {\varvec{\Gamma}}_{t}^{\alpha } + {\text{error}}_{t}^{\alpha } ,} \\ \end{array}$$

(5)

where error ^α _t is referred to as the idiosyncratic error of the model, $\lambda_{\alpha }$ is a ‘factor load’ coefficient that measures how much the cycle reacts to general shocks, which affect all the quantiles of the distribution. ${\varvec{\Gamma}}_{t}^{\alpha } = \lambda^{\alpha } \varvec{F}_{t}$ is the common component of the model, that is shocks that impact all the quantiles. If we define $\varvec{C}_{t} = \left( {c_{t}^{{\alpha_{1} }} ,c_{t}^{{\alpha_{2} }} , \ldots ,c_{t}^{{\alpha_{m} }} } \right)^{\prime }$ and ${\varvec{\Lambda}} = \left( {\lambda^{{\alpha_{1} }} , \ldots ,\lambda^{{\alpha_{m} }} } \right)^{\prime }$, where $\alpha_{j} , \,\,j = 1, \ldots m$ represents specific quantile indexed by j (and therefore 0 < α_j < 1), we have, in matrix form:

$$\begin{array}{*{20}c} {\varvec{C}_{t} } \\ {\left( {m \times 1} \right)} \\ \end{array} = \begin{array}{*{20}c} {{\varvec{\Lambda}} \varvec{F}_{t} } \\ {\left( {m \times r} \right)\left( {r \times 1} \right)} \\ \end{array} + \begin{array}{*{20}c} {{\text{error}}_{t} ,} \\ {\left( {m \times 1} \right)} \\ \end{array}$$

(6)

where ${\text{error}}_{t} = \left( {{\text{error}}_{t}^{{\alpha_{1} }} ,{\text{error}}_{t}^{{\alpha_{2} }} , \ldots ,{\text{error}}_{t}^{{\alpha_{m} }} } \right)^{\prime }$. Note that, without loss of generality, even when the model specifies a static relationship between c ^α _t and $\varvec{F}_{t}$, $\varvec{F}_{t}$ can be considered a dynamic vector process. If $\varvec{F}_{t}$ and $\varvec{X}_{t}$ are jointly stationary, then $\varvec{F}_{t}$ evolves according to a vector autoregression (VAR) process, as follows:

$$\varvec{A}\left( L \right)\varvec{F}_{t} = \varvec{u}_{t} ,$$

(7)

where $\varvec{A}\left( L \right)$ is a polynomial of the lag operator. This model is referred to in the literature as a dynamic factor model if $\varvec{F}_{t}$ includes primitive factors and their lags or as a static factor model if it accounts only for the primitive factors (Bai and Ng 2008).

The factors in Eq. 5 can be estimated using principal components (PC) or singular value decompositions (SVD), both methods allowing the estimation of the factors and the factor loads.

Identification issues arise owing to the fact that $\varvec{F}$ and ${\varvec{\Lambda}}$ are clearly not separately identifiable. For any arbitrary (r × r) invertible matrix $\varvec{H}$ we have that:

$$\varvec{F}{\mathbf{\varLambda^{\prime}}} = \varvec{FHH}^{ - 1} {\mathbf{\varLambda^{\prime}}} = \varvec{F}^{*} {\varvec{\Lambda}}^{'*} ,$$

(8)

where $\varvec{F}^{\varvec{*}} = \varvec{FH}$ and ${\varvec{\Lambda}}^{*} = {\varvec{\Lambda}}\varvec{H}^{ - 1}$. In this case, the factor model is observationally equivalent to $C = F^{*} \varLambda^{'*} + error$. Therefore, r² restrictions are required to uniquely fix $\varvec{F}$ and ${\varvec{\Lambda}}$ (Bai and Wang 2012). Notice that the estimation of the factors when using either PC or SVD imposes the normalization that $\frac{{\varLambda^{\prime}\varLambda }}{M} = I_{r}$ and $\varvec{F^{\prime}F}$ be diagonal, which are sufficient to guarantee identification (up to a column sign change). We follow this approach here.

1.2 Further Considerations When Dealing with Stochastic Versus Deterministic Trends

When working with time series, such as mortality rates or, as in this case, temporal quantiles of the life table survivorship function, the trends and cycles in the data must be modeled accurately in order to determine whether the time dynamics of the system respond to stochastic or deterministic trend components (or, alternatively, whether the processes are stationary in levels and, thus, do not house any trend at all).

The way of approaching each case differs considerably. For example, when the system is stationary in levels (that is, there is not a deterministic, nor a stochastic trend in the data), it is possible to project future patterns using traditional time-series ARMA-models (Enders 2010) or traditional principal components analysis (Stock and Watson 2002). In this case, the shocks to the system, for example, mortality reductions due to specific improvements in health treatments, vaccines, medical facilities, etc. would lack a permanent effect on mortality rates. Indeed, the effects of such shocks disappear after several periods, perhaps with some level of persistency, but in the long run, they do not modify the level of mortality rates.

However, if the series are trend-stationary, they have to be detrended before the estimation of causal or forecasting models. In this case, a simple deterministic model would suffice to describe the trend and thus, the projection of future patterns should focus on forecasting the cycles of the series, around the deterministic trend.

Finally, if the series are difference-stationary, traditional analysis is only valid after checking for cointegration (Engle and Granger 1987; Johansen 1988) or after differentiating each series, as many times as is required, to achieve stationarity. Cointegration refers to a situation in which two or more time series (mortality rates or survivorship quantile series) shared the same stochastic trend. That is, the effects of shocks that affect the dynamics of the series do not disappear and, moreover, they are the same for all the series.

In pragmatic terms, the first step to take is to check for unit roots in the data. Traditional and augmented unit root tests proposed by Dickey and Fuller (1979) and Said and Dickey (1984), respectively, are employed in this study, to determine whether the series present evidence of unit roots.

It is especially important to compare with the case in which the series are well described by a temporal trend that faces unexpected random breaks. In this case, traditional unit root tests suffer a very inconvenient lack of power and might easily conclude in favor of a unit root, even when the process is better described by a linear trend with one or several breaks. This is especially important in the present context, because, on the one hand, under the unit root hypothesis, any shock affecting the quantiles of the survivorship distribution would have a permanent effect on the trajectory of such quantiles. On the other hand, if the unit root hypothesis is rejected, the effects of a shock would disappear, around the deterministic trend, creating what we could label as mortality cycles. That is, cyclical patterns of increasing and reduction in mortality quantiles, around the deterministic trend. Under a deterministic trend model with a break, we could identify which shocks to the system produce a permanent effect, in terms of mortality reduction, and which of them disappear in few years. The former will be associated with the break dates, while the latter will describe the stationary cycles around the quantile-trend.

1.3 Testing Procedure

We first perform the test against the alternative hypothesis of an intercept. That is, assuming in the alternative hypothesis that the quantiles fluctuate around a constant mean, which would be the case of underlying constant mortality rates across age categories. As expected, the null of a unit root cannot be rejected in this case (that is, we reject the constant mortality hypothesis) regardless of the quantile under consideration. This result is not surprising, because of the clearly increasing dynamics during the sample of every series, which is evident in Fig. 3.

At this point, we had to construct the test with an alternative that included a time-trend. In this case, we found that for some of the quantiles, that is, those above the 45th percentile for females and those between the 30th and 40th percentiles for males, the null of a unit root must be rejected. This means that the processes within these ranges are trend-stationary and that they should not be modeled as if they contained a stochastic trend. This is an important finding. It means that the best model describing the data, within the aforementioned categories, is a simple deterministic linear model. In this case, given the constant slope of the straight line, no signs of increasing or decreasing pace in the process of longevity extension are found. It does not mean that mortality is constant at these categories, but rather that the rate of the mortality shift.

For quantiles lying outside the aforementioned range, it is unclear as to whether they are described by a unit root process or by a deterministic trend with a break. Table 1 reports the values of the corrected statistics and the corrected critical values, considering a break in both the null and the alternative, for the IO model described in Sect. 2.2. We only report the results of the tests for those quantiles for which we failed to reject the null of a unit root (see Table 2); thus, the blanks in this table correspond to the cases for which a simple linear model suffices and, therefore, we do not need to test for the IO model. In the other cases we do need to consider the presence of structural breaks in the trend. A consideration of this possibility shows that the null of a unit root is rejected at every quantile for females (with the sole exception of the 25th percentile).

Table 1 Unit root tests (t), critical values (cv) and fraction of the sample where the structural break is presented (θ) for the innovative outlier model

Table 2 Traditional unit root tests (t) for lifespan quantiles and critical values (cv)

In the case of men, the null hypothesis of a unit root is rejected at the 10th and 15th percentiles, at a confidence of 95%, and the 20th, 25th, and 95th percentiles, at a confidence of 90%. Percentiles lying between the 45th and 90th levels for males could, alternatively, be modeled via a stochastic trend. However, in this case, additional specification tests are needed before we might conclude in favor of the unit root hypothesis (i.e., tests allowing for a greater number of breaks, and different functional forms of the deterministic trends). Therefore, we prefer to model all the quantiles using the broken trend model, because it proves to be more appropriate in most cases.

Once again we are facing an interesting finding. The series of quantiles in almost all the cases are better described by a linear model with a one-time change in slope, than by a unit root process (stochastic trend). What is means is that, indeed, there are signs of acceleration or deceleration (depending on the sign of the rotation) in the rate of improvements for these quantiles, but such changes are one-time breaks during the whole sample. Once these big shocks have impacted mortality rates, subsequent shocks lack a permanent or cumulative effect on the quantile-trends of the lifespan distribution.

At this point, we need to identify the periods in which the breaks in the linear trends occurred and estimate the sign of such rotations. Given the results above, the break dates are determined endogenously and they are reported in Table 3. In this table, we calculated the break date for all the quantiles, even for which they are not statistically significant, in the sake of completeness.

Table 3 Unit root tests for the innovative outlier model