## Abstract

van Raalte et al. (2023) alerted demographers to the potential dangers of calculating cohort measures from the “diagonals” of gridded age–period (AP) data. In the case of cohort fertility, however, a minor change to the estimation procedure can mitigate the trend and cohort size biases that the authors identify. With an appropriate algorithm, researchers can estimate cohort fertility indices from AP data quite well.

## Introduction

which I will call the *AP estimator*. They compared $F1935,\u2009...\u2009,\u2009\u2009F1982$ to the Japanese cohort values published by the HFD. For this example, they showed that proportional errors $Fc\u2212\Phi c\Phi c$ range from −2.1% to +4.1%, that $Fc$ systematically overestimates cohort fertility when $\Phi c$ is decreasing from one cohort to the next (and underestimates when $\Phi c$ is increasing), and that the errors can be magnified by size differences between adjacent birth cohorts.

In this commentary, I make two points about their cautionary fertility example:

Raw HFD input data for Japan are in age–period format. So target $\Phi c$ values for Japan were

*also*estimated from a grid of Lexis squares and $[fxt]$ values, but with the multistep procedure described in the HFD Methods Protocol (HFD 2023). Errors in their example measure failure to match another algorithm, rather than failure to match true cohort values.There is a simple variant of the AP estimator that has a stronger demographic rationale and that better approximates the HFD algorithm. This alternative procedure also has less bias and smaller errors when tested against true $\Phi c$ values calculated from Lexis triangle data.

As a consequence, estimating cohort fertility from age–period data is less dangerous than implied in van Raalte et al.’s (2023) cautionary note.

## Alternative AP Diagonalization

Formula (2) looks reasonable, because it is a straightforward discretization of the integral formula for cohort fertility on a Lexis rate surface $f(x,t)$ with continuous ages and times. In comparing (1) and (2), we see that the AP estimator simply plugs in $fx,c+x$ to approximate the cohort–age rate $\varphi xc$. However, with discrete 1 × 1 AP squares, a cohort passes through a single-year age interval over two calendar years, not one.

Figure 1 illustrates this with an example. The fertility rate of 1980-born women between exact ages 20 and 21 depends on births and exposure from parts of two AP cells: [20,2000] and [20,2001]. The AP estimate $F1980$ calculated from (2) includes data from only the first of the two relevant calendar years. This causes the backward-looking temporal bias noted by the authors.

*AP2 estimator*for completed cohort fertility by summing over ages:

$F\u02dcc$ will not exactly equal $\Phi c$, for several reasons (such as changing fertility rates within one-year age intervals and cohort size differences), but it will typically be a very good approximation. Most importantly, it is not as vulnerable to temporal bias as the AP estimator $Fc$, because it is centered on the age–cohort parallelograms of interest.

## Empirical Comparisons

Figure 2 compares the AP estimator from Eq. (2) to the AP2 estimator from Eq. (5). The top left panel reproduces the cautionary Japanese example in the research note and adds the AP2 estimates as red dots connected by line segments.

The temporal centering of the AP2 estimator greatly reduces the bias that concerned the authors. Over the 1955–1970 cohorts, for example, the AP2 estimator does not have a notable positive bias. In fact, it comes much closer than the AP estimator to matching the HFD estimates. Similarly, the AP2 estimator avoids the AP underestimates for the cohorts born after 1975. As a bonus, the errors associated with the unusually small 1966 “Fire-Horse” cohort are much smaller with the temporally centered AP2 estimates.

The top right panel of Figure 2 shows the cumulative distribution of differences between the HFD estimates of Japanese cohort fertility and the two alternatives. Large errors are much less common with the AP2 procedure, and the mean error for the AP2 estimator is close to zero (+0.1%).

The bottom panels of Figure 2 show the equivalent results for France, a country for which the raw HFD data include year of mother's birth. For French females, cohort fertility rates are observed rather than estimated. Thus, in these bottom panels we compare AP and AP2 estimates to the true cohort completed fertility levels. Results are very similar to those for Japan: the AP2 estimates do not have the temporal bias of the AP estimates when there are strong trends across cohorts, they are typically smaller than AP errors in absolute value, and the mean AP2 error is very close to zero (+0.2%).

## Conclusion

The van Raalte et al. (2023) research note demonstrated that treating AP diagonal estimates as if they were true cohort measures can be problematic. It is important to carefully consider the dangers of this procedure. In the specific context of cohort fertility, however, the dangers are not as large as their research note suggests. Temporally centering the estimates by averaging two diagonals eliminates most of the bias problems that they highlight and greatly mitigates the cohort-size effects. In the Japan case, the centered estimates match the HFD cohort allocation procedure very well. In the France case, they match the actual cohort data very well. With proper caution, demographers can safely “diagonalize” AP fertility rates to learn about cohorts.

## Supplementary Material

R code and data for replicating the calculations in this commentary are available at https://github.com/schmert/cohort-diagonals.