A Cohort Model of Fertility Postponement

Abstract

We introduce a new formal model in which demographic behavior such as fertility is postponed by differing amounts depending only on cohort membership. The cohort-based model shows the effects of cohort shifts on period fertility measures and provides an accompanying tempo adjustment to determine the period fertility that would have occurred without postponement. Cohort-based postponement spans multiple periods and produces “fertility momentum,” with implications for future fertility rates. We illustrate several methods for model estimation and apply the model to fertility in several countries. We also compare the fit of period-based and cohort-based shift models to the recent Dutch fertility surface, showing how cohort- and period-based postponement can occur simultaneously.

Appendix

In this Appendix, we present methods for deriving the adjustment factors $1+S^{\prime }$ or $(1+S^{\prime })/(1-R^{\prime })$ from data. We also explain the inherent difficulty in distinguishing period-based postponement from cohort-based postponement. If shifts are purely period-driven, as modeled in Eq. (1), then the adjustment factor is $1/(1-R^{\prime }(t))$, and this quantity can be calculated from changes in the period mean age at birth. If shifts are purely cohort-driven, as in Eq. (2), then the adjustment factor is $1+S^{\prime }(c)$, which can be calculated from changes in the (truncated) mean age at birth for cohorts, as shown below. If shifts are a combination of period and cohort factors (Eq. (3)), the necessary adjustment factor is $(1+S^{\prime })/(1-R^{\prime })$. We show, in the second subsection of this Appendix, that the ratio of the derivatives of period and cohort fertility with respect to age gives an approximation of the adjustment factor for the combined model. We do not need to estimate either the shift function S or the baseline schedule f ₀ because TFR^† is calculated from the observed fertility schedule f.

For both the cohort-shift model and the combined model, it is necessary to control for the influence of changes in the period quantum q. This is achieved by choosing an easily identifiable feature that indicates quantum. We employ the height of the mode for this purpose, but other features could also work. For example, when studying mortality, one might choose the minimum value of the hazards. Our “quantum-normalized fertility rates,” $\tilde {f} (a,t)$, are calculated as follows. In each period t there is an age (or ages) with the peak fertility rate, and we denote this rate by m(t). We can easily calculate m(t) from observed data as Maximum _a {f(a,t)}. We then define the quantum-normalized rate as

$$\tilde{f} (a,t):=\frac {f(a,t)}{m(t)}. $$

Let m ₀ denote the peak value of the baseline schedule f ₀. Under the assumptions of purely cohort shifts, m(t)=m ₀ q(t), and the quantum-normalized rates are

$$ \tilde{f} (a,t)=\frac 1 {m_{0}} f_{0}(a-S(t-a)). $$

(7)

Under the assumptions of combined period and cohort shifts (Eq. (3)), $m(t)=m_{0}q(t)(1-R^{\prime }(t))$, and the quantum-normalized rates are

$$ \tilde{f} (a,t)=\frac 1 {m_{0}} f_{0}(a-S(t-a)-R(t)). $$

(8)

Note that although one could also employ the quantum-normalized rates when estimating the adjustment factor for the period-shift model, this will have no impact because raising or lowering the overall level of fertility will not change the mean age at birth.

Estimation Using (Truncated) Mean Age at Birth

The BF adjustment uses changes in period mean age at birth to estimate the factor $R^{\prime }$. If cohort c has completed its years of fertility, we can use the change in the cohort quantum-adjusted mean age at birth to estimate $S^{\prime }(c)$. For cohorts that have not yet completed the fertile years, we can estimate $S^{\prime }(c)$ using the change in mean age at birth truncated to the latest year of data.

Our formula depends on several quantities. Let μ(c) be the quantum-adjusted mean age at birth for cohort c as of latest available age; that is,

$$ \upmu(c)=\frac{{{\int}^{h}_{l}} x \tilde{f} (x,c+x)\ dx}{{{\int}^{h}_{l}} \tilde{f} (x,c+x)\ dx}, $$

(9)

where l is the lowest age of available fertility data for cohort c, and h is the highest age of available fertility data for cohort c. Because fertility data come in discrete form, we use an interpolating function to calculate integrals of $\tilde {f}$. Let v(a,c) be the proportion of cohort c’s truncated fertility that occurs at age a calculated from the quantum-adjusted schedules; that is,

$$ v(a,c)=\frac{ \tilde{f} (a,c+a)}{{{\int}^{h}_{l}} \tilde{f} (x,c+x)\ dx}. $$

(10)

Let ${\upmu }^{\prime }(c)$ be the derivative of μ(c). In practice this can be estimated using

$$\frac{\upmu(c+1)-\upmu(c-1)}2,$$

provided the same l and h are used for cohorts c+1 and c−1.

Use Eq. (7) to replace $\tilde {f} (a,t)$ with

$$\frac 1{m_{0}}f_{0}(a-S(t-a))$$

in Eq. (9). We get

$$\upmu(c) =\frac{{{\int}^{h}_{l}} x f_{0}(x-S(c))\ dx}{{{\int}^{h}_{l}} f_{0}(x-S(c))\ dx}.$$

Because l and h are constants, differentiating this new expression for μ(c) with respect to c and using the rule $(p/g)'=p^{\prime }/g-(p/g) (g^{\prime }/g)$ gives us

$$ {\upmu}^{\prime}(c)=\frac{-S^{\prime}(c){{\int}^{h}_{l}} x f_{0}^{\prime}(x-S(c))dx }{{{\int}^{h}_{l}} f_{0}(x-S(c)) dx} - \upmu(c) \frac{-S^{\prime}(c) {{\int}^{h}_{l}} f_{0}^{\prime}(x-S(c)) dx }{{{\int}^{h}_{l}} f_{0}(x-S(c)) dx}. $$

(11)

The left-hand fraction in Eq. (11) can be evaluated via integration by parts as

$$S^{\prime}(c)\frac{ l f_{0}(l-S(c))-hf_{0}(h-S(c))+{{\int}^{h}_{l}} f_{0}(x-S(c))\ dx }{{{\int}^{h}_{l}} f_{0}(x-S(c))\ dx} $$

$$= S^{\prime}(c)\cdot \Big(lv(l,c)-hv(h,c)+1\Big). $$

The right-hand fraction in Eq. (11) is

$$-\upmu(c)S^{\prime}(c)\frac{f_{0}(h-S(c))-f_{0}(l-S(c))}{{{\int}^{h}_{l}} f_{0}(x-S(c)) dx}=-S^{\prime}(c)\upmu(c)\Big(v(h,c)-v(l,c)\Big). $$

Subtracting this second fraction from the first gives us

$$ \begin{array}{rcl} {\upmu}^{\prime}(c)&=&S^{\prime}(c)\cdot \Big(lv(l,c)-hv(h,c)+1\Big) + S^{\prime}(c) \upmu(c)\Big(v(h,c)-v(l,c)\Big) \\ &&=S^{\prime}(c)\cdot \Big(v(l,c)(l-\upmu(c))- v(h,c)(h-\upmu(c))+1 \Big), \end{array} $$

and solving this for $S^{\prime }(c)$ yields this estimate of $S^{\prime }(c)$:

$$ S^{\prime}(c)=\frac{{\upmu}^{\prime}(c)}{1+v(h,c) (\upmu(c)-h)+v(l,c) (l-\upmu(c))}. $$

(12)

Estimation Using First Derivatives of Period and Cohort Fertility

For the combined model of cohort and period postponement given in Eq. (3), the adjustment factor needed to recover q(t) from the observed fertility rates is $(1+S^{\prime })/(1-R^{\prime })$. In this section, we show how to estimate this factor using a ratio of partial derivatives. Because these derivatives tend to be very small, this estimation technique is much more volatile than the method using mean age at birth presented in the previous section.

In the absence of postponement, the derivatives of period fertility and cohort fertility with respect to age should be the same. With postponement, this ratio of derivatives measures the degree to which age has been compressed in a period; thus, multiplying fertility rates by this ratio gives us the appropriate quantity to include in our adjusted total fertility rate. In mathematical terms, this means that

$$ \frac{1+S^{\prime}(t-a)}{1-R^{\prime}(t)} =\frac {\tilde{f}_{a}(a,t)}{\tilde{f}_{a}(a,t)+\tilde{f}_{t}(a,t)} =\frac{\frac{d}{da} \tilde{f} (a,t)}{\frac{d}{da} \tilde{f} (a,c+a)}. $$

(13)

Note that because both derivatives in this ratio can be zero at ages that correspond to peak levels of fertility and also at ages with no fertility, those ages cannot be used to calculate the adjustment factor.¹⁰

To see why the first equality in Eq. (13) holds, notice that by Eq. (7), $\tilde {f} (a,t)= f_{0}(a-R(t)-S(t-a))/m_{0}$. Differentiating gives us

$$\tilde{f}_{a}(a,t)= f_{0}^{\prime}(a-R(t)-S(t-a))(1+S^{\prime}(t-a))/ m_{0},$$

and

$$\tilde{f}_{t}(a,t)= -f_{0}^{\prime}(a-R(t)-S(t-a))(S^{\prime}(t-a)+R^{\prime}(t))/m_{0}.$$

Although Eq. (13) suffers from some instability near the ages at peak fertility, it has the advantage of being agnostic toward the origin of shifts; that is, if shifts are purely cohort-based or purely period-based, this equation still approximates the correct adjustment factor. In contrast, Eq. (12) is applicable only if shifts can be modeled entirely as cohort-based; likewise, $1/(1-{\upmu }^{\prime }(t))$ provides the appropriate adjustment factor only if shifts can be modeled as entirely period-based.

Distinguishing Period Shifts From Cohort Shifts

The combined model of fertility postponement (Eq. (3)) allows for concurrent cohort-based and period-based shifts, denoted by S and R, respectively. It is natural to wonder whether either of these two factors dominates. In this section, we show that up to linear terms, it is not possible to distinguish between cohort and period shifts from observed data. Nonetheless, it is still possible to estimate the tempo-adjusted TFR without specifically separating cohort and period components, as demonstrated in the previous subsection of this Appendix.

We start with the combined postponement model of Eq. (3):

$$f(a,t)=q(t)(1-R^{\prime}(t))f_{0}(a-R(t)-S(t-a)).$$

Expand R and S via Taylor series as R(t)=r ₀+r ₁ t plus terms of degree 2 and higher, and S(c)=s ₀+s ₁ c plus terms of degree 2 and higher. We may replace R(t), S(c), and f ₀ without changing any values of f(a,t) or q(t) as follows:

$$\tilde{R}(t):= \displaystyle \frac {R(t)-r_{0}-r_{1}t}{1-r_{1}},\ \ \ \ \tilde{S} (c):=\displaystyle \frac {S(c)+r_{0}+r_{1}c}{1-r_{1}},$$

$$\text{and} \ \ \ \tilde{f}_{0}(a):=(1-r_{1})f_{0}(a-r_{1}a).$$

Alternatively, we may replace R(t), S(c), and f ₀ with

$${\check R}(t):=\displaystyle \frac {R(t)+s_{0}+s_{1}t}{1+s_{1}},\ \ \ \ {\check S}(c):=\displaystyle \frac { S(c)-s_{0}-s_{1}c}{1+s_{1}},$$

$$\text{and} \ \ \ {\check f}_{0}(a):=(1+s_{1})f_{0}(a+s_{1}a),$$

and again f and q remain unchanged at all ages and times.

As an example, suppose that shifts are linear and purely period-based, so that S(c)=0 and R(t)=r ₀+r ₁ t. This scenario is indistinguishable from a linear cohort shift with

$$R(t)=0, \ \ S(c)=\frac{r_{0}+r_{1}c}{1-r_{1}} \ \ \text{and} \ \ \tilde{f}_{0}(a)=(1-r_{1})f_{0}(a(1-r_{1})).$$

In this case, the BF period-based tempo adjustment and our cohort-based tempo adjustment both produce the same result because $\displaystyle \frac 1 {1-R^{\prime }(t)}$ and $1+\tilde {S}^{\prime }$ are both equal to 1/(1−r ₁). This shows that when the period schedule shifts at a rate of r ₁, the cohort schedule shifts at a rate of r ₁/(1−r ₁), which is the result obtained by Zeng and Land (2002). Similarly, linear cohort shifts can be reinterpreted as linear period shifts; thus, if the cohort schedule shifts at a rate of r ₁, the period schedule shifts at a rate of r ₁/(1+r ₁), and the same tempo-adjusted TFR is obtained from either adjustment procedure.

Because the linear terms in R and S are interchangeable, there can be no definitive way to disentangle cohorts and periods. This unavoidable ambiguity between R and S suggests that neither the magnitude nor the direction of the shifts is intrinsic to the cohort or period approaches. This inherent undecidability between period and cohort makes efforts such as those by Bongaarts and Sobotka (2012) a considerable challenge.