4.1 Polynomial age functions

For the Taylor series to terminate in a finite number of terms, we require that $\frac{d^{\,j}g}{gy^{\,j}} = 0, \; \forall j > M$ and therefore that g(y) must be a polynomial in y of order M.

Sketch of Proof. Take g(y), a general polynomial of order M, and expand as a function of x and t. This can then be regrouped into an equivalent form that corresponds to the AP terms in the model, in order to see how g(y) can be absorbed into the AP structure

\begin{align*}g(y) &= \sum_{n\,=\,0}^M a_n y^n \\[3pt]\Rightarrow g(t-x) &= \sum_{n\,=\,0}^M a_n (t-x)^n \\[3pt]&= \sum_{n\,=\,0}^M a_n \sum_{m\,=\,0}^n {\left(\begin{array}{c} n\\[3pt] m \end{array}\right)} t^m (\!-x)^{n-m} \\[3pt]&= \sum_{n\,=\,0}^M a_n \left[ (\!-x)^n + \sum_{m\,=\,1}^n {\left(\begin{array}{c} n\\[3pt] m \end{array}\right)} t^m (\!-x)^{n-m} \right] \\[3pt]&= \sum_{n\,=\,0}^M a_n (\!-x)^n + \sum_{n=1}^M \sum_{l\,=\,0}^{n-1} a_n {\left(\begin{array}{c} n\\[3pt] l \end{array}\right)} t^{n-l} (\!-x)^l \\[3pt]&= \sum_{n\,=\,0}^M a_n (\!-x)^n + \sum_{l\,=\,0}^{M-1} (\!-x)^l \sum_{n=l+1}^{M} a_n {\left(\begin{array}{c} n\\[3pt] l \end{array}\right)} t^{n-l} \\[3pt]&= \sum_{n\,=\,0}^M a_n (\!-x)^n + \sum_{l\,=\,0}^{M-1} (\!-1)^l f^{(l)}(x) \sum_{n=l+1}^{M} a_n {\left(\begin{array}{c} n\\[3pt] l \end{array}\right)} t^{n-l}\\[3pt]&= a(x) + \sum_{l\,=\,0}^{M-1} f^{(l)}(x) k^{(l)}(t)\end{align*}

If there are age functions in the model of the form $f^{(j)}(x) = x^j$ of $j = 0, 1, \ldots M-1$ , the expression above corresponds to equation (9) with $a(x) = \sum_{n\,=\,0}^M a_n (\!-x)^n$ and $k^{(j)}(t) = (\!-1)^j \sum_{n\,=\,j\,+\,1}^{M} a_n {\left(\begin{array}{c} n\\[3pt] j \end{array}\right)} t^{n-j}$ . More generally, we only require that the age functions span the first $M-1$ polynomials, because these are equivalent to a model with $f^{(j)}(x) = x^j$ such as that in the derivation above.

We can think of the transformation as expanding the polynomial g(y) into terms in x and t, grouping these and then combining them with the appropriate AP terms. A model with age functions spanning the first $M-1$ polynomials therefore has an additional $M+1$ degrees of freedom (represented by the coefficients, $a_n$ , of the general polynomial) which do not affect the fit to the data. This is similar to the analysis in Wilmoth (Reference Wilmoth1990), which argues that higher order polynomial trends in the cohort parameters will cause identifiability problems in a mortality model if sufficient AP terms of suitable form exist within the model. These additional degrees of freedom mean that we need to impose an additional $M+1$ identifiability constraints, which assign the $M+1$ unidentifiable polynomial trends between the different AP and cohort terms in the model.

The simplest example of this is the transformation of the classic APC model described in section3. This has a single parametric age function $f(x) = 1$ which spans the polynomials to order 0. The model will then allow first-order polynomials (i.e. linear terms) to be added to the cohort parameters with offsets made to the static life function and the period term without changing the fitted mortality rates. These are exactly the invariant transformations described in equations (5) and (6). Consequently, we impose two additional identifiability constraints for the cohort parameters in the model to identify their level and linear trend.

4.1.1 The Plat models

In Plat (Reference Plat2009), two new APC mortality models were introduced. These can be writtenFootnote ¹⁵

(11) \begin{align}\ln\!(\mu_{x,t}) &= \alpha_x + \kappa^{(1)}_t + (x-\bar{x})\kappa^{(2)}_t + (\bar{x}-x)^+ \kappa^{(3)}_t + \gamma_{t\,-\,x} \end{align}

(12) \begin{align}\ln\!(\mu_{x,t}) &= \alpha_x + \kappa^{(1)}_t + (x-\bar{x})\kappa^{(2)}_t + \gamma_{t\,-\,x} \qquad\qquad \qquad\ \end{align}

The second of these models was introduced as a simplification of the first, with the expectation that it would be more suitable for modelling mortality at high ages. We call the model in equation (11) the “Plat model” and the model in equation (12) the “reduced Plat model” for this reasonFootnote ¹⁶.

The first point to note is that both the Plat and reduced Plat models nest the classic APC model, and therefore the invariant transformations in equations (4), (5) and (6) are also applicable for both models.

The second point to note is that these models also nest simple AP mortality modelsFootnote ¹⁷, and therefore the results of Hunt and Blake (Reference Hunt and Blake2020b) are still applicable. This means that the “locations” of the period functions are undefined and need to be identified by imposing a constraint on their levels. Usually, this is of the form

\begin{align*}\sum_t \kappa^{(i)}_t = 0\end{align*}

These invariant transformations were noted by Plat (Reference Plat2009) and used to impose suitable identifiability constraints.

However, the third point to note is that both of these models have age functions $f^{(1)}(x) = 1$ and $f^{(2)}(x) = (x-\bar{x})$ which span the polynomials to linear order. Using the result of Theorem 1, we should be able to find a transformation of the parameters which adds a quadratic polynomial in y to the cohort parameters but leaves the fitted mortality rates unchanged. Indeed, we find that the transformation

(13) \begin{align}\{\hat{\alpha}_x, \hat{\kappa}^{(1)}_t , \hat{\kappa}^{(2)}_t, \hat{\gamma}_y\} &= \{\alpha_x - d(x-\bar{x})^2,\notag \\&\,\,\,\quad \kappa^{(1)}_t - d(t-\bar{t})^2, \kappa^{(2)}_t + 2d(t-\bar{t}), \gamma_y + d(y-\bar{y})^2 \}\end{align}

leaves the fitted mortality rates unchanged for both the Plat and reduced Plat models. We say that these models have unidentifiable quadratic trends, which have to be manually allocated between the different parameters via identifiability constraints.

Hence, we require three identifiability constraints on the cohort parameters in the Plat and reduced Plat models, i.e., to apportion the level, linear trend and quadratic trend between the different AP and cohort terms, plus identifiability constraints on the levels of the period functions. This means that for full identification of the models, we require an additional identifiability constraint to those discussed in Plat (Reference Plat2009).

If the model user fails to allocate the quadratic trend between the different terms via an additional identifiability constraint, then the fitting algorithm will make an apportionment in order to achieve convergence. However, this apportionment will not be based on any particular desired demographic significance and will depend on the specific details of fitting algorithm, such as the starting parameter values used. To illustrate, instead of removing quadratic trends from the cohort parameters and apportioning them to the AP terms, the fitting algorithm may split any quadratic trends between the cohort parameters and the AP terms, giving values of $\gamma_y$ with an apparent quadratic trend in y. Not only is this contrary to our desired demographic significance, it can make comparing parameters across data sets difficult due to the presence or absence of quadratic trends which do not depend on the data.

In addition, a failure to fully identify the model can lead to inefficient fitting algorithms, which take a long time to converge to a solution, as discussed in Hunt and Villegas (Reference Hunt and Villegas2015). Furthermore, they can also give parameter estimates which are not robust to small changes in the data (e.g. an additional year of data), since such changes can cause the fitting algorithm to abruptly change the allocation of the unidentifiable trends. For these reasons, it is very important to ensure that the APC mortality models we use are fully identified by imposing sufficient identifiability constraints to uniquely estimate all the parameters in the model.

Following the same approach as used for the classic APC model, we might choose to impose the constraints in section 3 and extend these to impose $\sum_y n_y (y-\bar{y})^2 \gamma_y = 0$ to remove quadratic trends in the cohort parameters and allocate them to the AP terms. However, as with the classic APC model, this choice is arbitrary and a different choice of constraints will make no difference to the fitted mortality rates, only to the interpretation we give to the parameters.

In section 3, we saw that the lack of identifiability of the linear trends in the model, due to the transformation in equation (6), was of practical as well as theoretical importance because linear trends were often observed in both the age and period terms. Similarly, the transformation in equation (13) is of practical importance when fitting the Plat model, because we usually see some curvature in $\alpha_x$ at high ages and also systematic departures from the linearity of the period functionsFootnote ¹⁸. These quadratic trends will, therefore, not be distinguishable from a quadratic trend in the cohort parameters in the Plat model. However, because the observed magnitude of such trends is typically smaller than the linear trends observed in the AP terms, failure to fully identify the quadratic trend in the data will typically have a lower, though still important, impact than a failure to identify the linear trend.

It is worth noting that the transformation in equation (13) does not treat the different period functions equally, i.e., a term which is quadratic in t is added to $\kappa^{(1)}_t$ , a term linear in t is added to $\kappa^{(2)}_t$ , whilst $\kappa^{(3)}_t$ is unchanged by the invariant transformation for the Plat model. However, this is true only for the particular definition of the age functions shown. To illustrate, instead of the Plat model in equation (11), we could instead have chosen an equivalent model of the form

(14) \begin{align}\ln\!(\mu_{x,t}) &= \alpha_x + \kappa^{(1)}_t + (x-\bar{x})^+\kappa^{(2)}_t + (\bar{x}-x)^+ \kappa^{(3)}_t + \gamma_{t\,-\,x}\end{align}

Such a model will trivially give the same fitted mortality rates as that in equation (11) and has the same number of parameters and so will have the same number of identifiability issues. However, the transformation corresponding to equation (13) for this model will now add terms linear in t to both $\kappa^{(2)}_t$ and $\kappa^{(3)}_t$ . Specifically, for this model, we have the invariant transformation

(15) \begin{align}\{\hat{\alpha}_x, \hat{\kappa}^{(1)}_t , \hat{\kappa}^{(2)}_t, \hat{\kappa}^{(3)}_t, \hat{\gamma}_y\} &= \{\alpha_x - d(x-\bar{x})^2, \kappa^{(1)}_t - d(t-\bar{t})^2, \notag \\&\,\,\,\quad \kappa^{(2)}_t - 2d(t-\bar{t}), \kappa^{(3)}_t + 2d(t-\bar{t}), \gamma + d(y-\bar{y})^2 \}\end{align}

in contrast to the transformation in equation (13). Specifically, we note that whilst the transformation in equation (13) did not involve $\kappa^{(3)}_t$ , the transformation in equation (15) does. The invariant transformations of the model are therefore specific to the age functions present and may be different in different models, even if those models give an equivalent fit to data.

4.2 Exponential and trigonometric age functions

The other case where equation (10) potentially yields invariant transformations of the parameters occurs when the derivatives of g(y) are cyclical with period $M \leq N$ .

Sketch of Proof. In order for the derivatives of g(y) to be cyclical with period M, we require

(16) \begin{align}\frac{d^Mg}{dy^M} &= Kg\end{align}

for some non-zero constant K. Substituting this into equation (10) and comparing with equation (9) give

\begin{align*}g(t-x) &= \sum_{j\,=\,0}^{M-1} \left.\frac{d^{\,j}g}{dy^{\,j}}\right|_{y=-x} \sum_{k\,=\,1}^\infty \frac{1}{(j+kM)!}t^{j+kM} \\&= \sum_{j\,=\,0}^{M-1} f^{(j)}(x) k(t)\end{align*}

This is of the form of equation (9) if we set $k(t) = \sum_{k\,=\,1}^\infty \frac{1}{(j+kM)!}t^{j+kM}$ and have M age functions $f^{(j)}(x) = \left.\frac{d^{\,j}g}{dy^{\,j}}\right|_{y=-x}$ present in the model. It is interesting to note, therefore, that transformations of this form do not involve the static age function, as there is no term in the Taylor expansion of $g(t-x)$ corresponding to a(x)Footnote ¹⁹.

Equation (16) has solutions of the form

\begin{align*}g(y) &= \sum_{i\,=\,1}^M\Re [a_i\; \exp(k_i y)]\end{align*}

where $\Re[z]$ is the real part of the expression z, and the $k_i$ are the M roots of the equation $k_i^M = K$ . In general, these roots will be complex, and therefore, g(y) will be exponential, trigonometric or a combination of the two. In addition

\begin{align*}f^{(j)}(x) &= \left.\frac{d^{\,j}g}{dy^{\,j}}\right|_{y=-x} \\&= \sum_{i\,=\,1}^M\Re [a_i k_i^j \; \exp(\!-k_i x)]\end{align*}

and so the age functions present in the model will also be exponential or trigonometric.

Exponential AP terms can be included in models constructed using the “general procedure” of Hunt and Blake (Reference Hunt and Blake2014), where they are typically used to explain infant mortality. As an example, consider a model of the form

(17) \begin{align}\eta_{x,t} &= \alpha_x + \kappa^{(1)}_t + e^{-\lambda x} \kappa^{(2)}_t + \gamma_{t\,-\,x}\end{align}

This is an extension of the “exponential” model of Hunt and Blake (Reference Hunt and Blake2020b), with an additional cohort term. We typically require $\lambda > 0 $ to give the age function the demographic significance of governing rates of mortality at low ages. This model will allow the parameters to be transformed using

(18) \begin{align}\{\hat{\alpha}_x, \hat{\kappa}^{(1)}_t, \hat{\kappa}^{(2)}_t, \hat{\gamma}_y\} &= \{\alpha_x, \kappa^{(1)}_t, \kappa^{(2)}_t - a\; e^{\lambda t}, \gamma_y + a \; e^{\lambda y} \}\end{align}

This means that exponential trends in time within the (transformed) data are not uniquely identifiable as either AP or cohort effectsFootnote ²⁰. This transformation gives us an extra degree of freedom in the model which could be used to impose an additional identifiability constraint.

In this case, however, the imposition of an identifiability constraint will be of little practical importance. In section 3, we said that in order to be practically important, the unidentifiable deterministic trends must be present in both the age and period dimensions of the transformed data. Exponential trends in the model parameters will typically correspond to super-exponential growth or decline in the observed mortality rates if either $\eta_{x,t} = \ln\!(\mu_{x,t})$ or $\eta_{x,t} = \text{logit}(q_{x,t})$ . Super-exponential growth in mortality rates is not typically observed. We therefore do not experience problems when fitting the model to data as a result of any failure to be able to assign uniquely such a trend to the either AP or the cohort terms.

As another example, consider a model with trigonometric age functions of the form

(19) \begin{align}\eta_{x,t} &= \alpha_x + \kappa^{(1)}_t + \cos (\theta x) \kappa^{(2)}_t + \sin (\theta x) \kappa^{(3)}_t + \gamma_{t\,-\,x}\end{align}

For this model, we can transform the parameters using

(20) \begin{align}\{\hat{\alpha}_x, \hat{\kappa}^{(1)}_t, \hat{\kappa}^{(2)}_t, \hat{\kappa}^{(3)}_t, \hat{\gamma}_y\} = \{& \alpha_x, \kappa^{(1)}_t, \notag \\&\kappa^{(2)}_t - a \; \cos (\theta t) - b \; \sin(\theta t), \notag \\& \kappa^{(3)}_t + a \; \sin (\theta t) + b \; \cos(\theta t), \notag \\&\gamma_y + a \; \cos(\theta y) + b \; \sin(\theta y) \}\end{align}

This means that periodic patterns are not uniquely identifiable as either AP or cohort effectsFootnote ²¹.

As with the exponential functions, the presence of unidentifiable trigonometric trends in the model will be of little practical importance. Whilst the (transformed) data often exhibit periodic behaviour in the cohort and period effects, it is rare to see periodic behaviour across agesFootnote ²². Again, we do not have the unidentifiable deterministic trends for the model in both the age and period dimensions and consequently do not experience practical difficulties when fitting the model to data as a result of any failure to be able to assign uniquely such trends to the either AP or the cohort terms.

4.3 Other age functions

Other parametric age functions do not admit any additional invariant transformations involving the cohort parameters, except in the case where they are actually redefined polynomials, exponentials or trigonometric functions. For instance, the third AP term in the Plat model did not generate any extra interactions with the cohort parameters, beyond those of the reduced Plat model. This simplifies the identifiability issues of more complex mortality models with different types of age functions, such as those produced by the “general procedure” of Hunt and Blake (Reference Hunt and Blake2014), compared with what would otherwise be necessary, were, for instance, only polynomial age functions to be used.

4.4 Summary

In summary, issues with the identifiability of APC models relate to functions of year of birth which can be decomposed into purely AP terms. However, this is only true in models where the age functions take specific parametric forms – namely, polynomial, exponential and trigonometric functions. In such models, certain deterministic trends cannot be uniquely allocated between the AP and cohort terms in the model and so require the imposition of arbitrary identifiability constraints in order to uniquely specify the modelFootnote ²³. This is summarised in the flow chart in Figure 1.

Figure 1 Flow chart of identifiability issues in APC models.

5.1 Projecting general APC models

Consider the case of projecting an APC mortality model, which has been fitted using data over the period [1, T] to give mortality rates at time $\tau > T$ . From equation (2), we could write this as

\begin{align*}\eta_{x,\tau} &= \alpha_x + \boldsymbol{\beta}^\top _x \boldsymbol{\kappa}_\tau + \gamma_{\tau-x}\end{align*}

If the model has identifiability issues, then the projected mortality rates should be unchanged under exactly the same invariant transformations as the fitted mortality rates were, i.e., if we have an invariant transformation of the form of equation (8), namely

\begin{align*}\hat{\alpha}_x &= \alpha_x - a(x) \\\boldsymbol{\hat{\beta}}_x &= \boldsymbol{\beta}_x \\\boldsymbol{\hat{\kappa}}_t &= \boldsymbol{\kappa}_t - {\textbf{\textit{k}}}(t) \\\hat{\gamma}_y &= \gamma_y + g(y)\end{align*}

where a(x), $k^{(i)}(t)$ and g(y) satisfy equation (9), in which case

\begin{align*}\eta_{x,\tau} &= \hat{\alpha}_x + \boldsymbol{\hat{\beta}}^\top _x \boldsymbol{\hat{\kappa}}_\tau + \hat{\gamma}_{\tau-x}\end{align*}

The projected $\boldsymbol{\kappa}_\tau$ (and potentially the $\gamma_{\tau-x}$ ) will be random variables, whose distribution is a function of the historical, fitted values, i.e., $\boldsymbol{\kappa}_\tau = P_\kappa(\tau;\,\{\boldsymbol{\kappa}\})$ and $\gamma_{y} = P_\gamma (y;\,\{ \gamma \})$ . We said previously that we should use the same method of projection for all sets of parameters as a first step to ensure that the projected mortality rates do not depend upon the identifiability constraints. However, for different identifiability constraints, these processes will be estimated from different sets of fitted parameters, e.g., if we use $P_\kappa(\tau;\,\{\boldsymbol{\kappa}\})$ to project the untransformed period parameters, we must use $P_\kappa(\tau;\,\{\boldsymbol{\hat{\kappa}}\})$ to project the transformed period parameters. If we combine this with the invariance of the projected mortality rates, we have

\begin{align*}\alpha_x + \boldsymbol{\beta}^\top _x P_\kappa(\tau;\,\{\boldsymbol{\kappa}\}) + P_\gamma (\tau-x;\,\{ \gamma \}) &= \hat{\alpha}_x + \boldsymbol{\hat{\beta}}^\top _x P_\kappa(\tau;\,\{\boldsymbol{\hat{\kappa}}\}) + P_\gamma (\tau-x;\,\{ \hat{\gamma} \}) \\ &= \alpha_x - a(x) + \boldsymbol{\beta}^\top _x P_\kappa(\tau;\,\{\boldsymbol{\kappa} - {\textbf{\textit{k}}}\}) + P_\gamma (\tau-x;\,\{ \gamma + g\})\\P_\gamma (\tau-x;\,\{ \gamma + g\}) - P_\gamma (\tau-x;\,\{ \gamma \}) &= a(x) + \boldsymbol{\beta}^\top _x \left(P_\kappa(\tau;\,\{\boldsymbol{\kappa}\}) - P_\kappa(\tau;\,\{\boldsymbol{\kappa} - {\textbf{\textit{k}}}\}) \right)\end{align*}

Using equation (9), we can eliminate a(x)

\begin{align*}P_\gamma (\tau-x;\,\{ \gamma + g\}) - P_\gamma (\tau-x;\,\{ \gamma \}) &= g(\tau-x) + \boldsymbol{\beta}^\top _x \left(P_\kappa(\tau;\,\{\boldsymbol{\kappa}\}) - P_\kappa(\tau;\,\{\boldsymbol{\kappa} - {\textbf{\textit{k}}}\}) - {\textbf{\textit{k}}}(\tau) \right)\end{align*}

In order for this to hold for all $\tau$ and x requires

(21) \begin{align}P_\kappa(\tau;\,\{\boldsymbol{\kappa} - {\textbf{\textit{k}}}\}) &= P_\kappa(\tau;\,\{\boldsymbol{\kappa}\}) - {\textbf{\textit{k}}}(\tau) \end{align}

(22) \begin{align}P_\gamma (y;\,\{ \gamma + g\}) &= P_\gamma(y;\,\{\gamma\}) + g(y)\end{align}

This means that we should obtain the same results if we project the transformed parameters as if we transform the projected parameters, i.e., the processes of projection and transformation are commutative. Consequently, we see that, in order for a projection method to be well identified under the invariant transformation, it needs to preserve the unidentifiable trends in the model, i.e., $P_\kappa$ must preserve the trends ${\textbf{\textit{k}}}(t)$ , and $P_\gamma$ must preserve the trend g(y). This also means that it does not matter in which order we perform the processes of projection and transformation, the distribution of the transformed parameters projected into the future will be identical to the distribution of the projected parameters which are then transformed.

In addition, since

\begin{align*}\mathbb{V}ar(\boldsymbol{\kappa}_\tau) &= \mathbb{V}ar(\boldsymbol{\kappa}_\tau - {\textbf{\textit{k}}}(\tau)) = \mathbb{V}ar(\hat{\boldsymbol{\kappa}}_t) \\\mathbb{V}ar(\gamma_y) &= \mathbb{V}ar(\gamma_y + g(y)) = \mathbb{V}ar(\hat{\gamma}_y)\end{align*}

we note that the uncertainty of the parameters around the trend at any point in time is identifiable and so does have a meaning independent of the identifiability constraints imposed. Therefore, we conclude that, while the deterministic trends may be unidentifiable and not meaningful, the variation around the trend is of genuine significance, since it is independent of the identifiability constraints. Therefore, this variation needs to be projected consistent with our demographic significance for the parameters and what has been observed in the historical data.

However, the time series processes selected via current practice often do not preserve the unidentifiable trends in the period and cohort parameters, as we shall now see using the classic APC model.

5.2 Projecting the classic APC model

It has long been known, at least since Osmond (Reference Osmond1985), that the lack of identifiability in the classic APC model has important consequences when making projections from the model. Different sets of arbitrary identifiability constraints are based on different allocations of the linear trends in the data between the age, period and cohort parameters. The outcome of current practice can therefore be influenced by the presence or absence of a linear trend in the fitted parameters, despite this being purely dependent upon the identifiability constraints chosen.

To illustrate this, we consider projecting the classic APC model fitted using four different sets of identifiability constraints. The fitted mortality rates given using these four sets of constraints are identical; however, the time series processes found by current practice differ which means that current practice would give different projected mortality rates in the four different cases. Consequently, these time series processes are not well identified.

We start by fitting the classic APC model to mortality data for the USA from Human Mortality Database (2014) for ages 50 to 100 and year 1950 to 2010. As discussed in section 3, a number of equally valid identifiability constraints can be imposed on this model, which give identical fitted mortality rates. We consider the following four sets of identifiability constraints:

• Case 1: $\sum_t \kappa_t = 0$ , $\sum_y n_y \gamma_y = \sum_{x,t} \gamma_{t\,-\,x} = 0$ and $\sum_y n_y \gamma_y (y-\bar{y}) = \sum_{x,t} \gamma_{t\,-\,x} ((t-\bar{t})-(x-\bar{x})) = 0$ . This was discussed in section 3 and restricts the cohort parameters to be zero on average and without any linear trends, consistent with our desired demographic significance for the cohort parameters.

• Case 2: $\sum_t \kappa_t = 0$ , $\sum_y \gamma_y = 0$ and $\sum_y \gamma_y (y-\bar{y}) = 0$ . These constraints impose the same demographic interpretation on the parameters, except that the averages are not weighted by the number of observations of each cohort.

• Case 3: $\sum_t \kappa_t = 0$ , $\sum_{x,t} \gamma_{t\,-\,x} = 0$ and $\sum_{x,t} \gamma_{t\,-\,x} (x-\bar{x}) = 0$ . This set of constraints is the same as imposed on the classic APC model in Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009), where it was written as imposing $\sum_x (\alpha_x - \frac{1}{T} \sum_t \eta_{x,t} )(x-\bar{x}) = 0$ , i.e., that the static age function, $\alpha_x$ , explains all the linearity across ages in the data.

• Case 4: $\sum_t \kappa_t = 0$ , $\sum_{x,t} \gamma_{t\,-\,x} = 0$ and $\sum_{x,t} \gamma_{t\,-\,x} (t-\bar{t}) = 0$ . Similar to Case 3, this set of constraints imposes that the period function, $\kappa_t$ , accounts for all of the linearity across years in the data.

The first thing to note is that all of these constraints were developed to give the cohort parameters the same demographic significance, i.e., that they should be centred on zero and the other functions in the model should capture any linear trends. Because of this, the fitted parameters in each case are very similar. However, they are not identical, unlike the fitted mortality rates. We therefore see that demographic significance, whilst helpful in selecting an appropriate set of identifiability constraints, does not specify a unique set of constraints to use. Model users with the same interpretation of the parameters can reasonably choose to impose different constraints and obtain different fitted parameters when using the same model with the same data. The fact that demographic significance is subjective and, in practice, different model users adopt a range of interpretations for the different parameters highlights the fact that we must take care to ensure that any conclusions regarding projected mortality rates are independent of the arbitrary choice of constraints made when fitting the model, and underscores the extent to which the identifiability constraints we choose is arbitrary.

Current practice is to take the fitted parameters and then determine which time series processes to use to project them. This may involve performing a Box–Jenkins analysis on the fitted parameters, as was done in Lee and Carter (Reference Lee and Carter1992) and Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein and Khalaf-Allah2011). Alternatively, current practice may appeal to the demographic significance assigned to the parameters, as in Plat (Reference Plat2009). Such an appeal might determine that the period function is non-stationary (as it is primarily responsible for the evolution of mortality) and, based on the discussion in Hunt and Blake (Reference Hunt and Blake2020c), that the cohort parameters are stationary around zero. It might therefore appear reasonable to chooseFootnote ²⁶ to use a random walk with drift process for $\kappa_t$ and an AR(1) (first order autoregressive) process for $\gamma_y$

(23) \begin{align}\kappa_t &= \kappa_{t-1} + \mu + \epsilon_t \end{align}

(24) \begin{align}\gamma_y &= \rho \gamma_{y-1} + \varepsilon_y \quad\end{align}

Table 1 shows the fitted parameters for the four cases above using these time series processes.

Table 1. Time series parameters for the period and cohort functions in the classic APC model fitted using different identifiability constraints.

For $\tau-x > 1950$ Footnote ²⁷, we find

(25) \begin{align}\mathbb{E} \eta_{x,\tau} &= \alpha_x + \kappa_{2010} + (\tau-2010)\mu + \rho^{\tau-x - 1950}\gamma_{1950}\end{align}

We can therefore see that, inserting the fitted time series parameters from Table 1 for the four different cases, we do not find the same expected values for the future mortality ratesFootnote ²⁸. This is shown in Figure 2. In addition, the variability of the projected parameters depends on $\sigma_\kappa$ , $\rho$ and $\sigma_\gamma$ . However, $\rho$ and $\sigma_\gamma$ differ between cases, meaning that the variability of projected mortality rates will also be different for the different cases. These differences in the distribution of projected mortality rates might be felt to be relatively small, although they will grow with projection time. However, the most important point is that the differences should not exist at all – the fitted mortality rates for the different cases were identical and so should be the distribution of the projected mortality rates. We therefore see that the time series processes used above to project the classic APC model are not well identified.

Figure 2 Projected $\mu_{60,t}$ using different sets of identifiability constraints.

5.3 Projecting general APC mortality models: revisited

From section 5.1, we note that we must use the same time series processes to project sets of parameters which give identical fitted mortality rates, i.e., if $P_\gamma (y;\,\{\gamma\})$ is a suitable process (with time series parameters estimated from the fitted cohort parameters, $\{\gamma_y\}$ ), then $P_\gamma (y;\,\{\hat{\gamma}\})$ is a suitable process, albeit with time series parameters estimated from the transformed cohort parameters, $\{\hat{\gamma}_y = \gamma_y + g(y)\}$ .

In practice, we usually describe our projection methods in terms of time series processes rather than projection functions. However, the two are equivalent, since the projection function is found by “solving” the difference equation form of the time series. For instance, the AR(1) process has the difference equation form in equation (24) but has solution

\begin{align*}P_\gamma(y;\,\{\gamma\}) &= \rho^{y-Y} \gamma_Y + \sum_{s\,=\,Y\,+\,1}^y \rho^{y-s} \varepsilon_s\end{align*}

where Y is the last year of birth for which we fitted the cohort parameters.

The general form of ARIMA difference equations for $\gamma_y$ can be written asFootnote ²⁹

(26) \begin{align}(1-L)^d \Phi(L) (\gamma_y - \Gamma(y)) &= \Psi(L) \varepsilon_y\end{align}

where L is the lag operator, d is the order of integration of the process, $\Phi$ and $\Psi$ are polynomials of order p and q governing the autoregressive and moving average parts of the process, respectivelyFootnote ³⁰, $\varepsilon_y$ are the innovations and $\Gamma(y)$ is a deterministic function of year of birth. Taking unconditional expectations (i.e. with no conditioning on previous lags of the process), we see that

\begin{align*}\mathbb{E} \left[\gamma_y - \Gamma(y)\right] &= 0 \quad \forall y\end{align*}

and that the function $\Gamma(y)$ represents the trend around which the cohort parameters vary.

The invariant transformation of the model in equation (9) adds a deterministic function – the unidentifiable trend g(y) – to the cohort parameters. However, this deterministic function must not change the error term, $\varepsilon_y$ , of a well-identified process and so

\begin{align*}\varepsilon_y &= (1-L)^d \Psi^{-1}(L)\Phi(L) (\gamma_y - \Gamma(y)) \\&= (1-L)^d \Psi^{-1}(L)\Phi(L) (\hat{\gamma}_y - \hat{\Gamma}(y)) \\&= (1-L)^d \Psi^{-1}(L)\Phi(L) (\gamma_y +g(y) - \hat{\Gamma}(y))\end{align*}

In order to ensure that the variation around the trend, given by the error term, remains unchanged by the invariant transformation, we require

\begin{align*}\hat{\Gamma}(y) &= \Gamma(y) + g(y)\end{align*}

In this case, the deterministic trend, $\Gamma(y)$ , has changed under the invariant transformation but not the variation around the trend.

We stated above that the time series processes being used for the parameters should be equally applicable for all sets of parameters which give the same fitted mortality rates. This implies that the form of the deterministic trends should be the same and, therefore, that $\hat{\Gamma}(y)$ is of the same form as $\Gamma(y)$ . This can only be true if $\hat{\Gamma}(y)$ , $\Gamma(y)$ and g(y) are all of the same form. For instance, if g(y) is a linear function of year of birth (as in the case of the classic APC model), then $\Gamma(y)$ and $\hat{\Gamma}(y)$ must also be linear functions of year of birth and so will not change form under the invariant transformations of the model.

If we solve equation (26), we see that

(27) \begin{align}\gamma_y = P_\gamma(y;\,\{\gamma\}) &= \frac{\Psi(L)}{(1-L)^d \Phi(L)} \varepsilon_y + \Gamma(y)\end{align}

In this form, it can also be seen that such time series processes preserve unidentified trends in the manner discussed in section 5.1

\begin{align*}\hat{\gamma}_y &= \gamma_y + g(y) \\&= \frac{\Psi(L)}{(1-L)^d \Phi(L)} \varepsilon_y + \Gamma(y) + g(y) \\&= \frac{\Psi(L)}{(1-L)^d \Phi(L)} \varepsilon_y + \hat{\Gamma}(y)\end{align*}

i.e., the projected parameters after applying the invariant transformation will have the same variation, $\frac{\Psi(L)}{(1-L)^d \Phi(L)} \varepsilon_y$ , but around a different deterministic trend, $\hat{\Gamma}(y)$ , compared with the original parameters projected using the same method. The use of the invariant transformations will not affect our measurement of any coefficients in $\Psi(L)$ or $\Psi(L)$ at the fitting stage. Thus, we also see that the two ways of looking at the projected parameters, namely, as time series processes and via projection functions, are equivalent.

As an example, consider the cohort parameters in the classic APC model. From section 3, we see that, in this model, the cohort parameters have an unidentified constant and linear trend, i.e. $g(y) = b + c(y-\bar{y})$ from equations (5) and (6). In section 5.2, we said that current practice might use an AR(1) process for the cohort parameters, which has ARIMA form

\begin{align*}(1-\rho L) \gamma_y &= \varepsilon_y\end{align*}

Comparing this with equation (26), we see that current practice assumes that $\Gamma(y) = 0$ , which is not of the same form as g(y) above. Therefore, the time series process changes form when using an alternative set of parameters $\hat{\gamma}_y = \gamma_y + g(y)$ in place of $\gamma_y$ ,

\begin{align*}(1-\rho L) \hat{\gamma}_y &= (1-\rho L) (\gamma_y +b+c(y-\bar{y})) \\&= (1-\rho L) \gamma_y + (1-\rho)(b+c(y-\bar{y})) + \rho c \\&= \varepsilon_y + (1-\rho)(b+c(y-\bar{y})) + \rho c \\&\neq \varepsilon_y\end{align*}

and therefore the process is not well identified.

When analysed in this form, however, a solution becomes immediately apparent: we need to introduce a linear function, $\Gamma(y) = \beta_0 + \beta_1 y$ , into the AR(1) process to ensure that the process is well identified, i.e.,

(28) \begin{align}(1-\rho L) (\gamma_y - \beta_0 - \beta_1 y) &= \varepsilon_y\end{align}

Using the alternative parameters $\hat{\gamma}_y$ would produce

\begin{align*}(1-\rho L) (\hat{\gamma}_y - \hat{\beta}_0 - \hat{\beta}_1 y) &= (1-\rho L) (\gamma_y + b + c(y-\bar{y})- \hat{\beta}_0 - \hat{\beta}_1 y)\\&= (1-\rho L) (\gamma_y - \beta_0 - \beta_1 y) \\&= \varepsilon_y\end{align*}

if $\hat{\beta_0} =\beta_0 - b - c\bar{y}$ and $\hat{\beta}_1 = \beta_1 - c$ . Therefore, the form of equation (28) does not change under the invariant transformations of the classic APC model, and we conclude that this time series process is well identified. Again, we also see that the variation around the linear trend, given by $\varepsilon_y$ , is unchanged by the invariant transformation, whilst the unidentifiable trend is affected by the invariant transformation.

The time series process in equation (28) has been suggested previously for the cohort parameters in Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009) where it was referred to as the “AR(1) process around a linear drift”. However, in Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009), it was not used for the classic APC model, nor was it selected for being well identified, but rather on the grounds of fitting the observed cohort parameters well.

The AR(1) around linear drift process is solved to give

\begin{align*}P_\gamma (y;\,\{ \gamma\}) &= \rho^{y-Y} (\gamma_Y - \beta_0 - \beta_1 Y) + \beta_0 + \beta_1 y +\sum_{s\,=\,Y\,+\,1}^y \rho^{y-s} \varepsilon_s\end{align*}

We can also verify, by substituting the forms for $\hat{\gamma}_y$ , $\hat{\beta}_0$ and $\hat{\beta}_1$ found above, that this process also satisfies the requirement of equation (22) in section 5.1, namely

\begin{align*}P_\gamma(y;\,\{\hat{\gamma}\}) &= P_\gamma(y;\,\{\gamma\}) +a + b(y-\bar{y})\end{align*}

Hence, projecting the transformed cohort parameters gives us the same results as transforming the projected cohort parameters.

Returning to the form of the time series process in equation (26), it is common to write this in an alternative, but equivalent form

(29) \begin{align} (1-L)^d \Phi(L) \gamma_y - (1-L)^d \Phi(L) \Gamma(y) &= \Psi(L) \varepsilon_y \notag \\(1-L)^d \Phi(L) \gamma_y &= \xi(y) + \Psi(L) \varepsilon_y\end{align}

where $\xi(y)$ is a deterministic function of y and $\Gamma(y)$ solves the difference equation

(30) \begin{align}(1-L)^d \Phi(L) \Gamma(y) &= \xi(y)\end{align}

In this form, $\xi(y)$ is often referred to as the “drift”. Knowing the form that $\Gamma(y)$ must take (i.e. the same form as g(y) from the unidentifiable trends in the model in equation (8)), we can therefore specify the correct form of $\xi(y)$ .

As an example of this, consider the classic APC model again, but, this time, consider the period parameters. We know from section 3 that the period parameters have an unidentified linear trend in much the same way as the cohort parameters, i.e., $k(t) = a - c(t-\bar{t})$ if we rewrite equations (4) and (6) using the notation of equation (9). Random walk processes are often used for the period parameters, i.e., we assume $d=1$ and $\Phi(L)=\Psi(L) = 1$ . It is then important to specify the correct form for the drift $\xi(t)$ . Based on similar arguments to the ones used above for the cohort parameters, we should look for time series processes of the form

\begin{align*}(1-L) (\kappa_t - \nu_0 - \nu_1 t) &= \epsilon_t\end{align*}

which has a linear trend $K(t) = \nu_0 + \nu_1 t$ . To obtain a well-identified time series of the form of equation (29), we need the drift, $\xi(t)$ , of the random walk to satisfy

\begin{align*}\xi(t) &= (1-L)( \nu_0 + \nu_1 t)\\&= \nu_0 + \nu_1 t - \nu_0 - \nu_1 (t-1)\\&= \nu_1\end{align*}

i.e., the drift is constant. This shows that the random walk with drift is well identified for the period parameters in the classic APC model.

We can also verify this directly, since

\begin{align*}\epsilon_t &= \kappa_t - \kappa_{t-1} - \mu \\&= \hat{\kappa}_t - a + c(t-\bar{t}) - \hat{\kappa}_{t-1} + a - c(t-1-\bar{t}) - \mu \\&= \hat{\kappa}_t - \hat{\kappa}_{t-1} - \hat{\mu}\end{align*}

if $\hat{\mu} = \mu - c$ . Thus, the transformed period parameters, $\hat{\kappa}_t$ , follow a random walk with drift if the original period parameters do. However, the value of the drift, which determines the unidentifiable linear trend, will change under the invariant transformation, although the innovations, $\epsilon_t$ , which determine the variability around this drift do not.

In summary, we have the following procedure for selecting a well-identified time series process for any specific APC mortality model.

1. 1. Determine the identifiability issues in the specific APC model by finding the unidentifiable deterministic trends for the parameters which cannot be assigned between the different AP and cohort terms in the specific model. This will need to be done prior to the fitting stage in order to fit the model robustly to data.

2. 2. Specify a time series process for the variation around these trends. This can either be done by analysing this variation using statistical techniques, or by selecting a process which accords with our demographic significance for the parameters. Doing so will set the form of $\Phi(L)$ and $\Psi(L)$ , which determine the stochastic structure of the ARIMA process.

3. 3. Specify the deterministic trends, $\Gamma(y)$ , in the time series process in equation (26), which will need to be of the same form as g(y). Equivalently, this can be achieved by finding a drift function, $\xi(y)$ , in the alternative form of the time series process in equation (29), with the requirement that $(1-L)^d \Phi(L) \Gamma(y) = \xi(y)$ .

It is important to recognise that this procedure works backwards from the variation around the trends in the parameters, which is independent of the identifiability constraints and then adds back in the unidentifiable trends which will depend upon the specific set of identifiability constraints we use when fitting the model. In this fashion, we can ensure that the projected parameters are both well identified and possess our desired demographic significance when specifying a suitable form for the time series process.

5.4 Projecting the classic APC model: revisited

In section 5.2, it was demonstrated that the current practice approach to selecting time series processes for the period and cohort parameters in the classic APC model yielded projections of mortality rates which depended upon arbitrary choices made when fitting the model. In section 5.3, we then showed that the issue in this case was not the use of the random walk with drift for the period parameters, but the selection of an AR(1) process, rather than an AR(1) process around a linear drift for the cohort parameters.

If we use the AR(1) around linear drift process for the cohort parameters for the four cases discussed in section 5.2, we obtain the time series parameters in Table 2.

Table 2. Time series parameters for different identifiability constraints.

As previously mentioned in section 5.2, $\rho$ and $\sigma_\gamma$ control the variation of projected cohort parameters. It is, consequently, important to see that these parameters do not change in the four different cases using the well-identified time series processes. The variability of projected mortality rates will be identical in each of the four cases. Using the AR(1) around linear drift process, we also find

(31) \begin{align}\mathbb{E} \eta_{x,\tau} &= \alpha_x + \kappa_{2010} + (\tau-2010)\mu \notag \\&\quad + \rho^{\tau-x - 1950}(\gamma_{1950} - \beta_0 - \beta_1\times 1950) + \beta_0 + \beta_1 \times (\tau-x)\end{align}

From the results of section 5.3, we can see that if we transform the parameters of the classic APC model using the transformation in equations (4), (5) and (6), and then project them using well-identified time series processes, we obtain

\begin{align*}\hat{\alpha}_x &= \alpha_x - a - b + c(x-\bar{x}) \\[3pt]\mathbb{E} \hat{\kappa}_\tau &= \hat{\kappa}_{2010} + \hat{\mu}(\tau-2010) \\[3pt]&= \kappa_{2010} + a - c(2010-\bar{t})+ (\mu-c)(\tau-2010)\\[3pt]&= \kappa_{2010} + a - c(\tau-\bar{t}) + \mu(\tau-2010)\end{align*}

\begin{align*}\mathbb{E} \hat{\gamma}_{\tau-x} &= \rho^{\tau-x - 1950}(\hat{\gamma}_{1950} - \hat{\beta}_0 - \hat{\beta}_1\times 1950) + \hat{\beta}_0 + \hat{\beta}_1 \times (\tau-x) \\[3pt]&= \rho^{\tau-x - 1950}(\gamma_{1950} + b + c(1950 - x -\bar{y}) - \beta_0 - b -c\bar{y}- (\beta_1 +c)\times 1950) \\[3pt]&\quad + \beta_0 + b + c\bar{y} + (\beta_1+c) \times (\tau-x) \\[3pt]&= \rho^{\tau-x - 1950}(\gamma_{1950} - \beta_0 - \beta_1\times 1950) \\[3pt]& \quad + \beta_0 + \beta_1 (\tau-x) + c(\tau-x-\bar{y})\end{align*}

Hence, the expectation of $\eta_{x,t}$ in equation (31), after applying the invariant transformations, becomes

\begin{align*}\mathbb{E} \hat{\eta}_{x,\tau} &= \hat{\alpha}_x + \hat{\kappa}_{2010} + (\tau-2010)\hat{\mu} \notag \\[3pt]&\quad + \rho^{\tau-x - 1950}(\hat{\gamma}_{1950} - \hat{\beta}_0 - \hat{\beta}_1\times 1950) + \hat{\beta}_0 + \hat{\beta}_1 \times (\tau-x) \\[3pt]&= \alpha_x - a - b + c(x-\bar{x}) + \kappa_{2010} + a - c(\tau-\bar{t}) + \mu(\tau-2010) \\[3pt]&\quad + \rho^{\tau-x - 1950}(\gamma_{1950} - \beta_0 - \beta_1\times 1950) \\[3pt]&\quad + \beta_0 + \beta_1 (\tau-x) + c(\tau-x-\bar{y}) \\[3pt]&= \alpha_x + \kappa_{2010} + (\tau-2010)\mu \notag \\[3pt]&\quad + \rho^{\tau-x - 1950}(\gamma_{1950} - \beta_0 - \beta_1\times 1950) + \beta_0 + \beta_1 \times (\tau-x) \\[3pt]&= \mathbb{E} \eta_{x,\tau}\end{align*}

We can therefore see how changes in the linear drift of the period functions between the different cases cancel with the changes in the linear drift in the cohort functions to give exactly the same expected projected mortality rates in all four casesFootnote ³¹. We, therefore, see in practice what was derived theoretically in section 5.3, namely that using a random walk with drift process for the period parameters and an AR(1) around linear drift process for the cohort parameters gives well-identified projections for the classic APC model, and so the projected mortality rates which do not depend upon the identifiability constraints imposed.

Projections using an AR(1) process around a linear drift might be felt to conflict with our desired demographic significance for the cohort parameters, i.e., that they should exhibit no long-term trends. However, demographic significance is subjective and so should not be used to override a greater concern that the projected mortality rates do not depend upon the arbitrary identifiability constraints. Fortunately, there are methods for obtaining well-identified projections of the cohort parameters which do conform to our desired demographic significance of trendlessness.

In order to lack trends, the drift coefficients of the process, $\beta_0$ and $\beta_1$ , should be zero. Looking again at Table 2, one might think that the values of $\beta_0$ and $\beta_1$ are quite small and therefore be tempted to test them statistically with a view to setting them to zero. This, however, would be a mistake. As shown in section 5.3, the values of $\beta_0$ and $\beta_1$ change under the invariant transformations of the classic APC model and, therefore, will depend upon the identifiability constraints chosen. Consequently, the results of any statistical analysis of their significance will also depend upon the arbitrary identifiability constraints, which is not desirable.

The reason that $\beta_0$ and $\beta_1$ are “small” is because we have imposed this via the identifiability constraints. All four sets of identifiability constraints were chosen to set the level of the cohort parameters to be around zero and to have no linear trends over the whole range of the data. Therefore, we would expect to find low values of $\beta_0$ and $\beta_1$ , which control the level and drift to which the process mean-reverts. We could have chosen other, equally reasonable constraints based on alternative subjective interpretations of the demographic significance of the period and cohort parameters which would have resulted in far larger values of $\beta_0$ and $\beta_1$ and given exactly the same fitted and projected mortality rates. We therefore see that whether or not these parameters are “small”, and consequently whether or not they pass a statistical test of their significance, is solely dependent upon the arbitrary identifiability constraints we have chosen.

The four cases in section 5.2 were motivated by the same desired demographic significance for the cohort parameters – that they should be centred around zero and not have any linear trends. However, the four different cases used four different interpretations of these subjective requirements and therefore arrived at four different interpretations of what it means to be centred around zero and trendless. These different interpretations resulted in the four different sets of identifiability constraints. Using an AR(1) around linear drift process to project the cohort functions introduces a fifth interpretation for the meaning of being centred around zero and having no linear drift, in this case, that the time series parameters $\beta_0$ and $\beta_1$ are equal to zero. Therefore, we could use another set of parameters with the identifiability constraints

• Case 5: $\sum_t \kappa_t = 0$ , $\beta_0 = 0$ and $\beta_1 = 0$

This set of constraints gives identical fitted and projected mortality rates to the other cases but gives projected cohort parameters which mean-revert around zero, which accords better with our demographic significance. However, the restrictions in Case 5 cannot be known at the time of fitting the model to data, since the appropriate time series process that will be used to project the cohort parameters cannot be known at that stage. To use this set of constraints, we need to do the following:

1. 1. fit the model to data, applying some convenient set of identifiability constraints which can be known in advance of analysing the time series structure of the parameters, e.g., those in Case 1;

2. 2. estimate values for $\beta_0$ and $\beta_1$ for these historical parameters by fitting the AR(1) around a linear drift process in equation (28) to them;

3. 3. use these estimated values for $\beta_0$ and $\beta_1$ in the transformations in equations (5) and (6) to obtain a new set of (equivalent) age, period and cohort parameters.

The period and cohort parameters for Case 5, compared with those for Case 1, are shown in Figure 3. Using the Case 5 parameters may appear unnatural as the cohort parameters in this case appear to possess a linear trend. However, when we project using the well-identified AR(1) around linear drift process, we find no linear drift in these parameters, merely mean reversion to a level of zero, which fits well with the demographic significance for the cohort parameters discussed in Hunt and Blake (Reference Hunt and Blake2020c).

Figure 3 Projecting the parameters of the classic APC model: Cases 1 and 5. (a) Period parameters $\kappa_t$ . (b) Cohort parameters $\gamma_y$ .

5.5 Projecting the Plat model

We will now use this analysis to specify a set of well-identified projection processes for the Plat model discussed in section 4.1.1. As described in that section, the invariant transformations of the model can be written in the form of equation (9) with

\begin{align*}\begin{array}{lll}\hat{\alpha}_x &= \alpha_x - a_1 - a_2 - a_3 - b + c(x-\bar{x}) - d(x-\bar{x})^2 & = \alpha_x - a(x)\\[3pt]\hat{\kappa}^{(1)}_t &= \kappa^{(1)}_t + a_1 - c(t-\bar{t}) - d(t-\bar{t})^2 & = \kappa^{(1)}_t - k^{(1)}(t)\\[3pt]\hat{\kappa}^{(2)}_t &= \kappa^{(2)}_t + a_2 + 2d(t-\bar{t}) & = \kappa^{(2)}_t - k^{(2)}(t)\\[3pt]\hat{\kappa}^{(3)}_t &= \kappa^{(3)}_t + a_3 & = \kappa^{(3)}_t - k^{(3)}(t)\\[3pt]\hat{\gamma}_y &= \gamma_y + b + c(y-\bar{y}) + d(y-\bar{y})^2 & = \gamma_y + g(y)\end{array}\end{align*}

by composing the transformations in equations (4) (for each period function), (5), (6) and (13).

Starting with the cohort parameters, we may wish to retain the demographic interpretation that they should be stationary and mean reverting and so wish to use an AR(1) structure. However, from the discussion in section 5.3 and the observation that g(y) is quadratic for the Plat model, we therefore require that $\Gamma(y)$ in equation (26) is quadratic. In order to give well-identified projections, we would therefore project the cohort parameters using an AR(1) around quadratic drift process, i.e.,

(32) \begin{align}(1-\rho L) (\gamma_y - \beta_0 - \beta_1 y - \beta_2 y^2) &= \varepsilon_y\end{align}

Simple insertion of $\hat{\gamma}_y = \gamma_y + g(y)$ into this shows that it does not change structure under the invariant transformation and so is well identified. In principal, we could then decide to switch to an equivalent set of parameters with the constraints $\beta_0 = \beta_1 = \beta_2 = 0$ in the same manner as for the classic APC model. This may be desirable as it gives projected cohort parameters which mean-revert around zero, in line with our demographic significance. In addition, when more complicated methods are used to project the cohort parameters, it might be felt to simplify the process of projectionFootnote ³².

For the period parameters, we may wish to use a random walk with drift structure as we did for the classic APC model on the demographic interpretation that the period functions should be non-stationary. This would be written as

(33) \begin{align}(1-L) \boldsymbol{\kappa}_t &= \boldsymbol{\xi}(t) + \boldsymbol{\epsilon}_t\end{align}

where $\boldsymbol{\kappa} = \begin{pmatrix} \kappa^{(1)}_t, & \kappa^{(2)}_t, & \kappa^{(3)}_t \end{pmatrix}^\top$ as discussed in section 2 and similarly for $\boldsymbol{\xi}(t)$ and $\boldsymbol{\epsilon}_t$ .

Using this notation, we can group the transformations of the period functions as

\begin{align*}\boldsymbol{\hat{\kappa}}_t &= \boldsymbol{\kappa}_t + \begin{pmatrix} a_1+c\bar{t}-d\bar{t}^2 \\[3pt] a_2 \\[3pt] a_3 \end{pmatrix} + \begin{pmatrix} -c+2d\bar{t} \\[3pt] 2d \\[3pt] 0 \end{pmatrix} t + \begin{pmatrix} -d \\[3pt] 0 \\[3pt] 0 \end{pmatrix} t^2 \\&= \boldsymbol{\kappa}_t + {\textbf{\textit{k}}}_0 + {\textbf{\textit{k}}}_1 t + {\textbf{\textit{k}}}_2 t^2\end{align*}

In section 5.3, we showed that in order to ensure identifiability, we needed

\begin{align*}\boldsymbol{\xi}(t) &= (1-L) ({\textbf{\textit{k}}}_0 + {\textbf{\textit{k}}}_1 t + {\textbf{\textit{k}}}_2 t^2)\\&= {\textbf{\textit{k}}}_0 + {\textbf{\textit{k}}}_1 t + {\textbf{\textit{k}}}_2 t^2 - {\textbf{\textit{k}}}_0 - {\textbf{\textit{k}}}_1 (t-1) + {\textbf{\textit{k}}}_2 (t-1)^2 \\&= {\textbf{\textit{k}}}_1 - {\textbf{\textit{k}}}_2 + 2{\textbf{\textit{k}}}_2 t \\&= \begin{pmatrix} -c+2d\bar{t}+d \\ 2d \\ 0 \end{pmatrix} + 2\begin{pmatrix} -d \\ 0 \\ 0 \end{pmatrix} t\end{align*}

Therefore, we see that, in order for the Plat model to have well-identified projections, we require a constant drift component for $\kappa^{(2)}_t$ (i.e. $\xi^{(2)}(t) = \mu^{(2)}_0$ , a constant) and a linear drift component for $\kappa^{(1)}_t$ (i.e. $\xi^{(1)}(t) = \mu^{(1)}_0 + \mu^{(1)}_1 t$ , a linear function of time). This can be written as

(34) \begin{align}\boldsymbol{\kappa}_t &= \boldsymbol{\kappa}_{t-1} + \mu X_t + \boldsymbol{\epsilon}_t\end{align}

where

\begin{align*}\mu = \begin{pmatrix} \mu_0^{(1)}\qquad & \mu^{(1)}_t \\[3pt] \mu^{(2)}_0\qquad & 0 \\[3pt] 0\qquad & 0 \end{pmatrix}\end{align*}

and $X_t = \begin{pmatrix} 1, & t \end{pmatrix}^\top$ . We can see that this form of the random walk with drift process extends naturally to allow for other unidentifiable trends by choosing the “trend” matrix, $X_t$ , and corresponding “drift” matrix, $\mu$ , appropriately. The need to use a random walk with linear drift is often overlooked, for instance in Plat (Reference Plat2009) and Börger et al. (Reference Börger, Fleischer and Kuksin2013) (who used a model which nests the reduced Plat model).

We also see that different drifts are required for different period functions in order to give well-identified projections of mortality rates. This runs counter to the desire to treat all the period functions the same, as discussed in Hunt and Blake (Reference Hunt and Blake2020b). However, using the same drifts for all the period functions can give projections which are not biologically reasonable. For example, allowing for a quadratic trend in $\kappa^{(3)}_t$ can result in apparent changes in trend which are inconsistent with the historical data. In Hunt and Blake (Reference Hunt and Blake2020b), we also found that we can treat different period functions differently in models with parametric age functions, because there were no invariant transformations of the model which could be used to interchange the AP terms. It may, therefore, be preferable to allow for different drifts in different period functions in the Plat (Reference Plat2009) model to obtain well-identified projected mortality rates which are also biologically reasonableFootnote ³³. We should, therefore, be prepared to override the desire to treat the period functions identically if the alternative is to put biological reasonableness at stake.

5.6 Summary

APC mortality models which have unidentifiable trends at the fitting stage require extra care when projected to ensure that the projections do not depend on the identifiability constraints chosen. In general, we find that the projection method used must preserve whatever trends were unidentifiable at the fitting stage. For example, the processes which were well identified for the classic APC model discussed in section 5.4 preserved linear trends, which were shown to be unidentifiable in section 3.

Such an approach generalises naturally for more complicated mortality models, such as the Plat model discussed in sections 4.1.1 and 5.5. However, models with higher order polynomial age functions have higher order unidentifiable trends (as shown in section 4.1) and so require projection processes which allow for these trends. This may cause problems for long-term projections.

For example, consider the model

(35) \begin{align}\eta_{x,t} &= \alpha_x + \kappa^{(1)}_t + (x-\bar{x}) \kappa^{(2)}_t + ((x-\bar{x})^2 - \sigma_x) \kappa^{(3)}_t + \gamma_{t\,-\,x}\end{align}

which extends model M7 of Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009) with a static age function (as was done in Haberman and Renshaw (Reference Haberman and Renshaw2011)). We can see that a model of this form possesses age functions which span the polynomials to quadratic order. From section 4.1, we know, without performing any additional analysis, that it has unidentifiable cubic trends in both the cohort parameters and $\kappa^{(1)}_t$ which will need to be allowed for in projection. However small they may be in the historical data, these cubic trends will eventually come to dominate the long-term evolution of mortality rates, potentially yielding projected mortality rates which lack biological reasonableness due to apparent changes in trend.

Consequently, it may be prudent to avoid unidentifiable cubic (and higher) order polynomial trends in an APC mortality model. Such trends arise when we use more complicated models with higher order polynomial age functions. It is therefore useful, when selecting such models, to have a larger “toolkit” of age functions for use in the models than simply extending existing models by using higher order polynomial terms. Hunt and Blake (Reference Hunt and Blake2014) proposed such a toolkit, which allows for more complicated mortality models that do not suffer from excessive identifiability issues and can give biologically reasonable, well-identified projections of mortality rates, as shown in Hunt and Blake (Reference Hunt and Blake2020a), (Reference Hunt and Blake2015).