## Abstract

From a population perspective, the trajectories of both the total fertility at successive time periods and the total fertility of successive birth cohorts are derived from the same array of age-specific fertility rates. This analysis uses the assumption of constant age-specific fertility proportions to derive new explicit relationships between period and cohort fertility. In short, period total fertility is approximately equal to the total fertility of the cohort born a generation earlier, with a modest additive adjustment. A simple relationship also links both period and cohort total fertility to ACF, the average fertility of the childbearing cohorts in a given year. Assuming that fertility levels follow a cubic curve, cohort values from the derived relationships are then compared to observed cohort fertility values for the United States in 1917–2019. Despite substantial violations of the constant proportional fertility assumption, the calculated values deviate from the observed values by an average of only 7–8%. Short-term projections suggest that U.S. cohort fertility will continue to decline.

## Introduction

Ryder's (1964) defining article on “translation” in the first issue of *Demography* cast period fertility fluctuations as distortions of the underlying cohort behavior. Ryder (1964, 1965) saw the cohort, that is, the group of persons born in the same period, as a key sociological concept because social change has different impacts on persons of different ages, and the consequences of those impacts persist. In the study of fertility, Ryder believed the cohort was theoretically central because the behavior of different cohorts reflects those different socioeconomic conditions, and empirically significant because cohort fertility is more stable than period fertility. That translation perspective has since been updated by Keilman (2001) and further elaborated in Zeng and Land (2002).

In contrast to that cohort-centric view, Ní Bhrolcháin (1992) offered a period-centered perspective. She argued that period changes were unambiguously the prime source of fertility variation, and “if there are cohort effects in twentieth-century developed-country fertility series, they are so subtle as to be extremely difficult to detect” (Ní Bhrolcháin 1992:662). The empirical review by Hobcraft et al. (1982) saw consistency in the age pattern of *period* fertility but not in the age pattern of cohort fertility, indicating a greater consistency across age in period behavior. That suggests that the greater stability in cohort fertility is simply structural, in that cohort fertility just averages the highs and lows of period experience.

Cohort fertility has a clear heuristic appeal: women have children, mostly one at a time, and their past childbearing is obviously relevant to their subsequent behavior. Still, people live day by day, and those period influences can be long-lasting. The cohort emphasis ignores the fact that period fertility determines cohort size. There is also the practical limitation that the cohorts of greatest interest—those in their peak reproductive years—are decades away from completing their childbearing, and there is no generally accepted way of completing their fertility (Bohk-Ewald et al. 2018). Two decades ago, the issue of period fertility “distortions” arose again, with the controversy over whether recently observed period fertility declines actually reflected a change in cohort fertility behavior (Bongaarts and Feeney 1998; Frejka and Calot 2001; Kim and Schoen 2000; Lesthaeghe and Willems 1999; Schoen 2004; Van Imhoff and Keilman 2000).

Translation from period to cohort fertility can potentially help resolve such methodological questions. Translation adjustments to period fertility measures were first proposed by Ryder (1964), and a further noteworthy adjustment was proposed by Bongaarts and Feeney (1998), although both of those procedures use strong assumptions to relate cohort and period fertility.

To be specific, Ryder (1964) explored expansions of fertility functions in terms of moments in order to translate between period and cohort measures. His main result was the linear (first moment) relationship

*T*+

*A*(

_{c}*T*)) = CFR(

*T*) [1 – Δ

*A*(

_{c}*T*)],

where TFR indicates a period total fertility rate, CFR a cohort total fertility rate, Δ a change or first derivative over time, *A _{c}* a cohort mean age of fertility, and

*T*the birth year of a cohort. A total fertility rate is the sum of the age-specific fertility rates of a period (or cohort). For clarity, time index

*T*denotes a cohort year of birth while

*t*denotes period (or year)

*t*. Thus TFR(

*T*+

*A*(

_{c}*T*)) represents the period TFR in the year that the cohort born in year

*T*is at its mean age of fertility.

Bongaarts and Feeney (1998) adopted a period perspective and proposed the alternative linear relationship

*t*) = TFR(

*t*) / [1 – Δ

*A*(

_{p}*t*)],

where TFR*(*t*) is the Bongaarts–Feeney adjusted TFR designed to eliminate period distortions, and *A _{p}* is the period mean age of fertility. Bongaarts and Feeney (1998) made their adjustment parity specific, but Eq. (2) captures the essence of their approach. They asserted that their TFR* better reflected the completed family size (i.e., cohort fertility) than did the observed period total fertility rates. Zeng and Land (2002:270) clearly expressed the assumptions underlying the Bongaarts and Feeney TFR*, that is, TFR* was the CFR of the hypothetical cohort whose fixed rate schedule shifted each reproductive year by a fixed amount. Thus TFR* rests on both a fixed fertility rate schedule and 30–35 years of constant upward (or downward) shifts in that schedule.

Such linear-based adjustments are at best approximations to the actual period/cohort relationship. The goal of this article is to further explore and quantify period/cohort relationships and, from that perspective, examine recent trends in U.S. period and cohort fertility.

## Period/Cohort Relationships Under Constant Proportional Fertility

### The Constant Proportional Fertility Model

To begin, assume a constant proportional age schedule of fertility rates in every year, while the level of fertility is free to vary from period to period. In the context of such a model, we can relate period and cohort fertility based on the moments of the constant age-specific fertility proportions and a known function that specifies the fertility level over time. By doing so, both period and cohort fertility can be expressed in terms of a few underlying parameters.

We build a model where the relationships between period and cohort fertility are stated explicitly. At time *t*, let the fertility rate at exact age *x* be *f*(*x*,*t*). Assume no mortality before age 45, and let the reproductive ages be from 15 through 44. Denote the fixed proportion of period fertility at age *x* by *c*(*x*), with ∫*c*(*x*) *dx* = 1. Unless otherwise specified, integrals range from age 15 to age 45. At time *t*, let those fixed fertility proportions be multiplied by fertility level function TFR(*t*), so time *t* fertility at age *x* is given by *f*(*x*,*t*) = TFR(*t*) *c*(*x*), with the sum of the time *t* age-specific fertility rates equal to TFR(*t*).

The constant proportional fertility assumption is somewhat strong, and the period mean age of fertility (*A _{p}*) has risen considerably in many places over the past several decades. Nonetheless, the Hobcraft et al. (1982) finding that the age pattern of fertility is quite stable provides a reasonable point of departure. In the absence of plausible constraints, period/cohort relationships would be virtually impossible to analyze. The stable population, the dominant model of formal demography, is based on constant age-specific fertility and mortality rates. Despite those very strong and unrealistic assumptions, the stable model has been very useful in analytical work and in estimating demographic measures from incomplete data (

*cf.*United Nations 1983). Unlike the Ryder and Bongaarts–Feeney analyses, constant fertility proportions allow fertility levels to vary freely over time.

The United Nations 2014 population projections used a modified constant fertility proportions approach, with a population's initial age-specific proportions moving to fixed ultimate proportions (Ševčíková et al. 2016:301–302). The recent comprehensive analyses of fertility projection and cohort completion by Bohk-Ewald et al. (2018) found that more complex projection methods do not necessarily do better than simpler methods. In particular, they found that the assumption of constant (“frozen”) rates “consistently outperforms most of the sophisticated and less sophisticated forecast methods” (Bohk-Ewald et al. 2018:9191). With TFR(*t*) able to vary freely and the *c*(*x*) specific to the case at hand, the constant fertility proportions assumption is flexible, demographically grounded, and far weaker than assumptions frequently employed in empirical demographic modeling. Furthermore, the primary objective here is to relate period and cohort fertility, not to improve techniques for population projection. Assuming constant fertility proportions allows new, explicit relationships to be derived that directly link the total fertility of a year to the completed fertility of a cohort. The best-performing methods in Bohk-Ewald et al. (2018), which use time series/extrapolations or Bayesian methods, do not provide such direct period/cohort relationships.

Now let us relate period and cohort fertility when TFR(*t*) is a known polynomial function. With CFR(*T*) the total fertility (or completed family size) of the cohort born at time *T*, we can write

where the integral ranges over the 30-year reproductive age span. Equation (3), our starting point, provides the basic relationship between period and cohort fertility.

### Relationships Under a Linear Fertility Trajectory

Explicit relationships can now be derived for specific fertility trajectories. When the fertility trajectory is linear, we can write

*t*) =

*R*+

*at*,

where parameter *R* is the level of fertility at time 0 and parameter *a* is the linear slope. Then, from Eq. (3), with integration from ages 15 to 45, we can write

where µ, the mean of the *c*(*x*) distribution, equals ∫ *x c*(*x*) *dx*. With a linear time trajectory, the completed family size of the cohort born in year *T* is *identical* to TFR(*T* + µ), the total fertility rate of the year (*T* + µ).

Figure 1 shows period and cohort age curves of fertility when *R* = 1 and *a* = 0.02. To simplify the calculations, the constant proportional age schedule of fertility is given by the parabola

*c*(

*x*) = (−3 / 20) +

*x*/ 75 –

*x*

^{2}/ 4500

between the ages of 15 and 45 and is zero at all other ages. The derivation of the curve is straightforward and described in section D of the online appendix and Appendix Supplement 1. The *c*(*x*) curve is zero at ages 15 and 45 and has an area of 1 between those ages. It is symmetric, with a mean at age 30, where *c*(30) = 0.05, a variance of 45, and zero skew. While it simplifies the typical age curve of fertility, that parabolic curve affords a reasonable depiction of actual behavior.

In Figure 1, both period (*t* = 30) and cohort (*T* = 0) total fertility are 1.6. The two curves cross over at exact age 30. Below age 30, period fertility is higher; after age 30, cohort fertility is higher. More specifically, the excess of period fertility over cohort fertility at age 30 − *k* (0 < *k* < 15) exactly equals the excess of cohort fertility over period fertility at age 30 + *k*. Section D in the online appendix gives the values underlying Figure 1 for five-year age-groups, showing that symmetry.

The mean ages of fertility are also significant functions, and their values can readily be found. With constant *c*(*x*) and a linear change in fertility, the period mean age of fertility (*A _{p}*) is always µ, while the cohort mean age,

*A*, is given by

_{c}*A*(

_{c}*T*) = µ +

*a*Var / CFR(

*T*) = µ +

*a*Var / TFR(

*T*+ µ),

where *A _{c}*(

*T*) = ∫

*x c*(

*x*) [

*R*+

*a*(

*t*+

*x*)]

*dx*and Var denotes the variance of the

*c*(

*x*) distribution, that is, ∫(

*x*− µ)

^{2}

*c*(

*x*)

*dx*. The full derivation of Eq. (7) is given in section A of the online appendix. Unlike the period mean age, the cohort mean age varies over time, differing from µ by a combination of the slope of the linear trajectory (

*a*), the dispersion of the age pattern of fertility (Var), and the total fertility of the cohort.

The Ryder (1964) linear translation relationship in Eq. (1) is consistent with Eq. (5), when the *cohort* mean age of fertility is used. Employing Eqs. (1), (4), (5), and (7), the *left* side of Eq. (1) can be written

To see that the right side of Eq. (1) also yields the result in Eq. (8), first find the change in *A _{c}*(

*T*), that is,

*dA*(

_{c}*T*) /

*dT*= −

*a*

^{2}Var / [

*R*+

*a*(

*T*+ µ)]

^{2}.

Using Eq. (5) and substituting the relationship in Eq. (9) in the *right* side of Eq. (1) yields

where the last equality in Eq. (10) equals that in Eq. (8).

It is easier to show that the Bongaarts–Feeney Eq. (2) is consistent with Eq. (5) for the *period* mean age of fertility**.** There is no adjustment, as µ is constant. Hence, at time *t* = *T* + µ, we have TFR*(*t*) = TFR(*t*) = TFR(*T* + µ) = CFR(*T*).

The consistency of results in the linear case between the Ryder and the Bongaarts–Feeney approaches on one hand and the constant *c*(*x*) approach on the other hand reinforces the plausibility of the constant proportional fertility assumption. Since the constant *c*(*x*) approach can be applied to nonlinear fertility trajectories, it extends those earlier approximations and opens the door to more general relationships.

### Relationships Under a Quadratic Fertility Trajectory

When period fertility follows a quadratic curve, we can write

*t*) =

*R*+

*at*+

*bt*

^{2}.

Then, integrating using Eqs. (3) and (11) gives

*T*) = TFR(

*T*+ µ) +

*b*Var

and

*A*(

_{c}*T*) = µ + {Var [

*a*+ 2

*b*(

*T*+ µ)]} / CFR(

*T*).

Section B of the online appendix shows the details of the derivation of Eqs. (12) and (13).

In the quadratic case, the difference between CFR(*T*) and TFR(*T* + µ) is a constant, the extent of nonlinearity in the TFR trajectory times the variance of the *c*(*x*) distribution. The difference between the period and cohort mean ages varies nonlinearly over time.

### Relationships Under a Cubic Fertility Trajectory

When fertility levels follow a cubic curve, let us write

*t*) =

*R*+

*at*+

*bt*

^{2}+

*dt*

^{3}.

Proceeding as before, section C of the online appendix shows that we then have

*T*) = TFR(

*T*+ µ) + Var [

*b*+ 3

*d*(

*T*+ µ)]

and

*A*(

_{c}*T*) = µ + Var {

*a*+ 2

*b*(

*T*+ µ) +3

*d*[(

*T*+ µ)

^{2}+ Var]} / CFR(

*T*).

When the level of fertility follows a cubic curve, the difference between CFR(*T*) and TFR(*T* + µ) varies linearly over time. If parameter *d* equals zero, that is, if the fertility trajectory is quadratic, Eqs. (15) and (16) reduce to Eqs. (12) and (13). A cubic curve is generally considered a reasonable approximation for most observed trajectories over the short to medium range (e.g., Hoem and Linnemann 1988). Miller's classic actuarial text on graduation goes so far as to state that “over a limited range, most regular series met with by the actuary may be closely approximated by a polynomial of the third degree” (Miller 1946:26).

The quadratic and cubic derivations provide new period/cohort relationships that go beyond the linear relationships in Ryder (1964) and Bongaarts and Feeney (1998). Those new relationships indicate that cohort fertility levels and mean ages are the period values for the time when the cohort is at the period mean age, adjusted *additively* for nonlinearity in the trajectory of period fertility and for the mean and variance of the fixed age pattern of fertility. Table 1 summarizes those period/cohort relationships for linear, quadratic, and cubic fertility trajectories.

### Relationships in Hypothetical Data With a Cubic Trajectory

Calculations with hypothetical data can illustrate the magnitude of those additive adjustments. Table 2 shows fertility levels and mean ages when fertility follows a cubic trajectory with either low or high fluctuations. The low-fluctuation pattern, roughly based on twentieth-century Swedish experience, has TFR varying from a high of 2.31 to a low of 1.54. The high-fluctuation pattern, somewhat similar to the United States in the twentieth century, has TFRs varying from 3.0 to 1.8.

With low fluctuations, the difference between CFR(*T*) and TFR(*T* + µ) is rather small, on the order of one hundredth of a child. That difference is precisely the adjustment to TFR(*T* + µ) in Eq. (15). Figure 2 shows period and cohort total fertility trajectories under both scenarios. The period and cohort curves are distinct, but quite similar. With high fluctuations, period/cohort differences are larger, but completed family sizes differ by only ±0.1 child.

The difference between cohort mean age *A _{c}* (

*T*) and period mean age µ is more substantial. With low fluctuations, it goes from nearly −0.5 to over +0.3, and with high fluctuations from −1.7 to +0.2. That difference is precisely the additive adjustment to µ in Eq. (16). In short, a linear trend assumption fails to capture the complexities of the cubic case, although there is still a considerable closeness between the values for the cohort born in year

*T*and the period

*T*+ µ.

## Average Cohort Fertility With Constant Proportional Fertility

Butz and Ward (1979), in examining the effects of timing on period and cohort fertility, introduced the average cohort fertility (ACF) concept. As a weighted average of the fertility of cohorts childbearing at time *t*, the ACF provides a period measure that offers insight into cohort behavior. Under the constant proportional fertility assumption, let us write

with *t* = *T* + µ and the integral spanning the ages 15 to 45. The definition in Eq. (17) differs from that of Butz and Ward (1979) in that the weights are the fraction of period (not cohort) fertility at each age.

Now assume that period total fertility varies cubically over time per Eq. (14), so cohort total fertility varies according to Eq. (15). Our period-weighted ACF can then be written

*t*) = ∫ [TFR(

*t*+ µ −

*x*) + Var (

*b*+ 3

*d*{

*t*+ µ −

*x*})]

*c*(

*x*)

*dx*.

As shown in section E of the online appendix, straightforward integration over the range of *c*(*x*) yields

*t*) = TFR(

*t*) +2 Var (

*b*+ 3

*dt*) = CFR(

*T*) + Var (

*b*+ 3

*dt*).

Equation (19) shows that under a regime of constant proportional fertility whose level varies cubically over time, there is a simple relationship between period-weighted ACF at any time and both period and cohort total fertility. From Eqs. (15) and (19), CFR(*T*) is the arithmetic mean of TFR(*t*) and ACF(*t*), that is,

*T*) = [TFR(

*t*) + ACF(

*t*)] / 2.

Equations (19) and (20) also hold when fertility varies quadratically, and if fertility changes linearly, that is, when *b* = *d* = 0, ACF(*t*) = TFR(*t*) = CFR(*T*) = CFR(*t* − µ). These ACF relationships are new and afford another approach to period/cohort translation and to inferring cohort behavior from period measures.

## Period/Cohort Relationships in American Fertility Data

Detailed fertility data for the United States go back to 1917, and those data provide an excellent source for examining the extent to which our model-based period/cohort relationships hold in an actual population over a long span of time. The data sources used are described in section F of the online appendix.

Table 3 shows the period TFR(*t*), the period *A _{p}* (

*t*), and the CFR(

*T*) for the years 1917 through 2019. Since the fertility data are by calendar year and the

*A*values are not integral, a simplifying procedure was used to determine the appropriate birth cohort year

_{p}*T*=

*t*−

*A*. Specifically, observed

_{p}*A*fractions of ±(1/3) were ignored. If the fractional part of

_{p}*A*(

_{p}*t*) was between 1/3 and 2/3, the fraction was assumed to be one half, and the CFR values for two years were averaged. For example, for an

*A*(

_{p}*t*) value of integral

*Z*plus one half, CFR(

*T*) was taken as ½[CFR(

*t*−

*Z*) + CFR(

*t*–

*Z*− 1)].

In the continuous time models of Table 1, those born at exact time *T* are at exact age *x* at exact time *T* + *x*. With discrete data, those born in year *T* all attain exact age *x* in year *T* + *x*, although at age *x* last birthday they live some person-years during year *T* + *x* + 1. To simplify matters, recognizing that single years of age and time are considered and that fractional mean ages of fertility are involved, the calculations assume that the behavior at age *x* of the cohort born in year *T* is captured by data for the year *T* + *x*.

Figure 3 shows the trajectories of TFR(*t*), CFR(*t* − *A _{p}*(

*t*)), and the cubic estimated CFR for the cohort attaining age

*A*(

_{p}*t*) in year

*t*. The maximum U.S. period TFR was 3.68 in 1957. Fertility then fell to 1.74 in 1976 before rising again to 2.12 in 2007. In the following 12 years, fertility fell again, reaching an all-time U.S. low of 1.705 in 2019. Cohort fertility peaked at 3.18 for the cohorts born in 1933–1934, before falling to 1.98 for the cohort born in 1954. Subsequently, the CFR rose, reaching 2.24 for women born in 1978. For the most recent seven years, the CFR declined, but was still 2.08 for the cohort of 1984.

Since 1965, or the cohorts of 1938–1939, CFR(*T*) has generally been greater than TFR(*t*). That reflects the rise in the period mean age of fertility. From 25.74 in 1974, *A _{p}* (

*t*) rose nearly monotonically to 29.62 in 2019, a rise of 3.88 years. Such a substantial increase in

*A*, and relatedly in the fertility rates at higher ages, directly counters the model assumption of constant proportional fertility and provides a stringent test of the robustness of the model.

_{p}The estimated CFR trajectory in Figure 3 was calculated under the cubic assumption using Eq. (15). For each year *t*, a cubic curve was fit to TFR values for years *t* − 15, *t* − 5, *t* + 5, and *t* + 15, and the parameters of that curve were used in Eq. (15). The 30-year period was chosen as it approximates both the length of a generation and the reproductive age span. Alternative specifications were considered, but differences were modest. The details are given in section G of the online appendix, and alternative estimates are shown in Appendix Table 1.

Figure 3 shows that the cubic estimated CFR tracks the observed CFR quite well, although it is a bit closer to the corresponding TFR. The largest errors are during the baby boom of the 1950s and the birth dearth of the 1970s, where the estimated CFR still outperforms the TFR. The robustness of the cubic CFR values to violations of the constant proportional fertility assumption is quite striking. Appendix Tables 1 and 2 show year-by-year and overall differences between the observed CFR and the different estimates. The linear or lagged TFR estimate was off by 0.196. On a base of 2, that is a percentage error of 9.8%. The average difference from the Eq. (15) cubic estimates was between 0.144 and 0.157, giving percentage errors of 7.2% to 7.8%.

Equations (14) and (15) relate period total fertility to cohort total fertility, but those relationships only partially constrain *age-specific* fertility, allowing many possible fertility schedules. One reasonable way to calculate age-specific rates is by a normal curve parameterization, which only requires knowledge of the mean and variance and is fully consistent with the derivation of Eq. (15). Approximating age-specific fertility by a normal curve ignores the typical right-handed skew of the fertility rates but is a frequent practice in demography (*cf.*Keyfitz 1977: chapter 6).

Figure 4 shows observed age-specific fertility and the fertility rates estimated from a normal curve parameterization for the United States for 1925, 1950, 1975, and 2000. (Details of the method used are in section I of the online appendix.) The fit of the estimates is quite good for 1975, and so-so for the other three years, where the skew in the observed rates is more pronounced. Thus plausible age-specific fertility rates can be estimated, even though the relationships derived here between the TFRs and the CFRs only loosely constrain the age curve of fertility.

To sum up, the model-based relationships of Table 1 perform reasonably well, even though the constant fertility proportions assumption is substantially violated. Alternative choices in operationalizing the model make little difference. Period fertility, especially when adjusted, consistently approximates the cohort fertility of U.S. women born 26–29 years earlier.

## Forward- and Back-Projecting Cohort Total Fertility

Now let us consider whether the relationships explored here can extend the CFR series, first further into the future and then to earlier cohorts. With *A _{p}* (2019) equal to 29.62 years, TFR(2019) corresponds to the birth cohort of 1989–1990, while Table 3 only provides CFRs through the birth cohort of 1984.

One way to further extend the CFR series is to complete the experience of all active cohorts. The study of Bohk-Ewald et al. (2018) indicates that simply assuming that the age-specific birth rates of the latest year continue to apply generally yields quite good results. Table 3 uses 2019 experience in that way but, to minimize error, only at ages above 35. Those ages account for only 19% of 2019 fertility, while ages 30 and above account for 48% of 2019 fertility. Thus the assumption that fertility above age 30 remains constant at 2019 levels is much stronger than the one used to calculate CFR values in Table 3.

A second way to extend the series is by the linear assumption of Eq. (5), which takes TFR(2019) alone as the estimate of CFR(1989–1990). From the relationships in Figure 3, that assumption is a bit crude.

A third possibility is to use the cubic trajectory assumption of Eq. (14) and estimate CFRs up to the cohort of 1989–1990 using Eq. (15). That way appears promising, but implementing it requires fitting cubic curves to determine parameters *b* and *d*. Formally, that can be done by using the last 30 years of TFR data, as described in section H of the online appendix. Substantively, however, there is no escaping the use of a cubic curve that extends the fertility series to 2034–2035, some 15 years beyond the last data point. Doing so places extraordinary reliance on the cubic assumption.

Table 4 presents results from those three approaches for the years 2005–2019.

Column 4—the Observed TFR based on fertility to at least age 35—gives the most reliable figures, but only goes to the cohort born in 1984. Comparing the figures in column 4 to the corresponding TFR indicates that the observed CFRs are from 0.10 to 0.24 higher (see column 5).

Column 6 shows that the cubic projected CFRs increase to a maximum of 2.076 for the cohort of 1979, and then decline steadily to 1.599 for the cohort of 1989–1990. For the years 2005–2013, column 7 shows how those projected CFRs relate to the observed CFRs in column 4. The cubic projected CFRs are uniformly higher, by 0.14 to 0.31, and for the last three years of the comparison are 0.3 larger. That suggests that the cubic curve may be falling too rapidly.

In sum, while each has its limitations, the three approaches can offer a plausible range for CFRs through the birth cohort of 1989–1990. Cohort fertility is not likely to fall below 1.6, the cubic projection figure. Taking the 2019 TFR, the 1989–1990 cohort will have a CFR of 1.7. However, the TFR series is about 0.2 below the observed series in column 3, so completed family size may be as high as 1.9. In any case, within the likely range of 1.6 to 1.9, the CFR for the 1989–1990 cohort will clearly be below replacement and very likely the lowest CFR in American history.

Similar issues arise in back projections to estimate the fertility of earlier cohorts (see section H of the online appendix for details). To extend the cubic approximation back to cohorts born in 1888–1889, the cubic curve based on the years 1917, 1927, 1937, and 1947 has to be extrapolated back to 1887. The results are shown in Table 5. The cubic back-projected CFRs are given in column 5 and show a largely steady decline from 4.137 for the 1888–1889 cohort to 2.614 for the 1903 cohort. The differences from the CFRs reported in Heuser (1976), shown in column 6, reveal that the back-projected figures are uniformly larger, with differences greater than 0.7 for the cohorts born from 1888–1889 through 1898. That very poor performance by the back projection reflects the dangers involved in extrapolating cubic curves. In contrast, column 7 shows that the lagged TFR values perform fairly well, with an average error of 0.18.

## Summary and Conclusions

The relationship between period and cohort fertility has been a focus of demographic interest since Ryder's pioneering 1964 paper. Here, a new approach to relating those two perspectives is developed, based on the assumption of constant age-specific fertility proportions. When the level of fertility follows a linear, quadratic, or cubic trajectory, explicit relationships, shown in Table 1, are found between period total fertility at time *t* and the total fertility of the cohort born at time *T*, where *T* = *t* − µ and µ is the constant period mean age of fertility. Analogous relationships are found for the period and cohort mean ages of fertility, and a simple relationship unites both period and cohort total fertility with the average fertility of a year's actively childbearing cohorts.

These new relationships between period and cohort total fertility are quite robust to departures from the assumed pattern of constant age-specific fertility proportions. That is evident when estimated cohort total fertility is compared to observed cohort total fertility for the United States in 1917–2019. U.S. fertility was at an all-time low of 1.705 in 2019, and the data show substantial departures from the assumed fixed proportional fertility. Nonetheless, the theoretically derived estimates had errors averaging only 7–8%.

Both period and cohort perspectives illuminate fertility behavior. Period total fertility is not a “distortion” of cohort total fertility, but another manifestation of the same array of age- and time-specific fertility rates. As women bear children over their life course in the light of their previous fertility, the heuristic appeal of the cohort concept is undeniable. Nonetheless, people live year by year, and those period conditions influence both short-term and long-term behavior. What emerges from the present analysis is a fuller appreciation of the close relationship between period and cohort fertility. The two perspectives are complementary, and both contribute to our understanding of population fertility.