Tempo and the TFR

One reading is that any measure of period fertility that is influenced by genuine timing effects is distorted by definition, and that it is a construct “period fertility” that is wrongly measured by the TFR. Some scholars appear to espouse a view close to this, seeing tempo effects, in and of themselves, as biasing to period measures. For example, the earlier quotation from Bongaarts and Feeney implies that timing change is by definition distorting (see also Bongaarts 2002, 2008; Bongaarts and Feeney 2000; Kohler and Philipov 2001; Zeng and Land 2002). This is a defensible position for which a case might, in principle, be made. But it would have to be put forward and justified explicitly because, as I will show in a later section, one can equally argue that genuine tempo effects are intrinsic to the fertility of a period.

A second way of seeing distortion in the TFR as measurement bias is in Norman Ryder’s sense: that period parameters are “distorted reflections of cohort behavior” and that the time series of period and cohort values do not coincide when fertility timing is changing (Ryder 1964:79, 1980). This implicit construal of bias/distortion—viewing period measures as erroneous indicators of real cohort values—is fairly common in the literature on tempo adjustment (Bongaarts 2008; Kohler and Ortega 2002a; Schoen 2004; Smallwood 2002b; Sobotka 2003; van Imhoff 2001; van Imhoff and Keilman 2000).

The conventional period TFR measures (real) cohort fertility accurately only if age-specific rates are either fixed or randomly distributed around a given period’s values. Such stability is rare. The TFR and analogous synthetic indicators are therefore unquestionably biased as estimates of real cohort parameters. This type of bias is exacerbated by timing change. Furthermore, it is present in empirical populations even when timing is constant. Thus, the period TFR is a distorted version of associated cohort values for two reasons: because fertility timing changes and because rates are not fixed.

A third interpretation is within a theoretical framework. In a theoretical population with fixed age-specific rates, the period TFR is an unbiased measure of cohort completed fertility, but it becomes biased as a cohort indicator when a theoretical tempo change is introduced. As just noted, the TFR in empirical populations is a biased indicator of cohort total fertility regardless of whether timing is changing. In most discussion of tempo adjustment, tempo change is treated as the exclusive cause of bias in period indices, a view that makes sense only if it refers to the relationship between the period and cohort TFR in theoretical rather than in empirical populations.

Adjustment for bias/distortion is proposed primarily for the period TFR and other synthetic indicators. A first question, then, is whether indicators of this kind are suitable as dependent variables when explaining period trends.³ I think not, for two reasons. First, the TFR is a single figure indicator, but period change is multidimensional rather than unidimensional and thus cannot be represented by a single figure. A summary indicator, however, may be adequate for analyses of the long run that aim to explain gross changes in level. Second, the TFR and analogous indicators are in synthetic cohort form, referring to the cumulative experience of a lifetime, and are therefore in an inappropriate metric to represent the phenomena of a single period (Ní Bhrolcháin 1992). As a result, they are unsuitable as dependent variables in any substantive model of the underlying process in its period aspect. As noted earlier, synthetic indicators can be treated as simple statistical summaries, and in that guise might be suitable as dependent variables for empirical as distinct from behavioral models. However, the problem remains that no single measure can accurately represent fertility in time series when trends differ between exposure categories.

The arguments in the preceding section apply to the TFR in all its forms: additive, multiplicative, adjusted, or unadjusted. Nevertheless, let us set aside objections to TFR-like indices and consider the issue of bias/distortion in the TFR when period fertility is the explanandum. Of the types of bias identified earlier, two are potentially relevant to trends in the period TFR as dependent variable: confounding (bias A) and definitional (bias B1). Bias of types B2 and B3 cannot be present because when we are trying to explain trends in period fertility, the dependent variable measures a period phenomenon and is not a cohort estimator.

Year-on-year movements in the TFR can be biased by compositional factors because of confounding between the exposure distribution and calendar time—a spurious timing effect (bias A). The solution here is not adjustment, which is designed to remove genuine timing effects, but to use indices either specific for parity and age, or for parity and duration for orders two and higher, or standardized for these factors. Such indicators, and measures summarizing them—period parity progression ratios, the period parity progression based TFR, and regression-standardized parity-specific rates (Feeney and Yu 1987; Hoem 1993; Ní Bhrolcháin 1987; Rallu and Toulemon 1994)—are free of confounding bias. But, occurrence-exposure rates of this type, and measures derived from them, are influenced by genuine tempo change and thus might be thought of as definitionally distorted (bias B1). That is a mistake. True tempo effects do not distort period indicators in the role of explanandum. Rather, genuine tempo effects are an integral component of the period fertility trends we have to explain, as are the changes in variance highlighted by Kohler and Philipov (2001). Explanation means explaining the whole of the change in period fertility, not just part of it. To remove tempo effects from period fertility as explanandum would denude it of an intrinsic and often substantial component of change (Ní Bhrolcháin 1992, 1994). The greater the timing shift in a period, the worse the impact of adjustment because a larger part of what is happening in a period, from an explanatory angle, is removed. Wachter (2005) sees tempo adjustment as a way of standardizing for tempo. I argue that a period fertility indicator that is standardized for tempo is misspecified, if the objective is to explain period trends. The issue is not at all comparable to age standardization in mortality analysis. In explaining variation in death rates, age structure effects are regarded as extraneous and not a target of explanation. In explaining change in period fertility, by contrast, identifying the causes of tempo change is part of the objective.

An extreme example, the Year of the Fire Horse in Japan, illustrates the argument. Japan saw a dip of 27% in the TFR in 1966, due to avoidance of births in a year that, according to popular astrology, would be an inauspicious year of birth for a girl. To explain the phenomenon, it would be absurd to use as the dependent variable a time series of tempo-adjusted TFRs. Any timing element is integral to the impact of folk belief on that year’s fertility. (Of course, if estimating the effect of the folk belief on cohort fertility, cohort outcomes would be the dependent variable.) In the case of the speed premium in Sweden, tempo is a large part of the period effect, and its precise detail allows a strong case to be made for a causal influence of legislation relating to maternity-pay provision on fertility (Andersson 1999; Hoem 1990; Ní Bhrolcháin and Dyson 2007). The same applies to less-pronounced period fluctuations. For example, accelerated childbearing was a sizeable ingredient of the baby boom of the late 1950s and 1960s in the United States, the United Kingdom, and several other developed societies (Butz and Ward 1979; Ní Bhrolcháin 1987; Ryder 1980). If part of the explanation of the baby boom is that postwar prosperity, full employment, and high wages gave rise to accelerated marriage and childbearing, that faster pace of family formation must be represented on the left hand side of the equation. Similarly, later childbearing is a central feature of what is to be explained in the period trends of the last few decades in developed societies.

An analogy may be useful. Consider a car traveling for a fixed duration of time. Its speed varies during the journey: rounding a sharp bend or going uphill, it slows down, while on the straight or downhill, it travels faster. Speed may vary also depending on terrain, traffic, the driver’s inclinations, and so on. The task of explaining period fertility trends is analogous to accounting for the sequence of speeds at which the car travels. Saying that a well-standardized period fertility indicator is distorted is like saying that a measure of the car’s speed at an arbitrarily chosen point in the journey, or when the car is changing speed, is mistaken. It may well give a biased estimate of average speed over the journey as a whole, but it gives an accurate account of the car’s speed at the point at which this was measured. If we think in terms of “underlying” speed or average speed during a journey, and whether and how it can be inferred from speed along the route, we are measuring something other than speed at a particular time-point. In addition, we would have to either construct models and make assumptions for the purpose, or investigate the properties of a large number of such journeys, to generate an empirical basis for the estimate.

The analogy is not perfect, but it helps to spotlight some key points. To adjust the measure of speed during periods of acceleration and deceleration would clearly misrepresent the recorded time sequence of speeds during the journey. Quantifying and explaining that sequence—comparable to measuring and explaining period fertility trends—is a different problem from measuring and explaining average speed or distance traveled during the journey—a task analogous to estimating cohort or longer-run fertility levels. Schoen (2004) has used the car analogy for a different purpose—to argue for the importance of cohort fertility—and assumes that the driver has an intended destination, although one that may alter during the journey. Here, the analogy is between the car’s trajectory and aggregate fertility movements, and no assumption is needed about intentions regarding destination, speed, or duration of the journey.

The argument thus far is that genuine timing effects are intrinsic to period fertility as dependent variable, but that does not rule out the disaggregation of period fertility into tempo and quantum components, each to be the focus of explanation. Separate estimates of period level and timing effects for explanatory purposes could be argued for if several conditions were to hold. These conditions are that in period mode: (1) the quantum of fertility and its timing are separable in a quantitative sense; (2) quantum and tempo measures reflect distinct aspects of the underlying behavioral process (when we are seeking a behavioral explanation rather than an empirical one); and (3) that quantum and tempo respond differently to change in social, economic, and other determining factors.

Are tempo and quantum separable in a quantitative sense? Indices can be specified that, on their face, represent the quantum and tempo aspects of period fertility (e.g., Bongaarts and Feeney 1998; Butz and Ward 1979; Foster 1990; Kohler and Ortega 2002a; Ryder 1980). However, although separable in theory, quantum and tempo tend to covary in practice. The most familiar evidence of this is the near-universal tendency of cohort fertility series to reflect the fluctuations in corresponding period series, but with a lesser amplitude.⁴ Time series presented by Schoen (2004) illustrate the point further. Schoen’s Table 1 gives a number of quantum indicators for the United States during 1917–2001: the conventional TFR, two versions of the Bongaarts and Feeney adjusted TFR, the Butz and Ward average completed fertility (ACF) measure,⁵ together with the mean age at childbearing and corresponding (true) cohort total fertility for part of the series. From these, the period timing indices associated with each quantum measure have been obtained. Corresponding to the ACF, we have the timing measure TFR / ACF (Butz and Ward 1979; Schoen 2004). Two specifications of a Bongaarts and Feeney timing index are calculated for each of the two versions of the BF TFR: additive (BF timing = TFR – TFR*) and ratio versions (BF timing = TFR / TFR*).

In all cases, the timing and quantum indicators are positively associated. The correlation between ACF and its associated timing index is .83 (1917–1997); and for the four versions of the Bongaarts and Feeney timing and quantum indices, the correlations are between .32 and .45 (1918–1997). On the ACF evidence, periods of faster timing are also periods of higher levels of fertility. This is true also of the Bongaarts and Feeney indices, although to a lesser extent, possibly because the TFR_adj indicator can be erratic (Schoen 2004; Smallwood 2002b). Overall, these figures demonstrate that although separable to some degree, period quantum and tempo are not independent (see also van Imhoff and Keilman 2000).

Are period quantum and tempo behaviorally distinct? Estimates of tempo and quantum effects such as those of Butz and Ward (1979), Ryder (1980), Foster (1990), and Bongaarts and Feeney (1998) are interesting. But, thus far, there have been no studies that investigate whether the mathematical constructs in these analyses correspond to real-world processes. The question is relevant at both micro and macro levels. At the individual level, the question is whether decisions about tempo and quantum are taken separately or jointly. At the aggregate level, whether the tempo/quantum divide corresponds to underlying processes is harder to determine, but it can be studied indirectly by investigating both the independence of tempo and quantum, and how far they are jointly or separately influenced by causal factors.

Ryder’s perspective on the subject is instructive. In a paper that is best known for having estimated the quantum and tempo of cohort fertility and for having analyzed them into their components, Ryder (1980) concluded that the two are not independent. In his view, quantum and tempo “are to some degree manifestations of the same underlying behavior,” and “we cannot, in principle, make a statistical separation of the tempo and quantum facets of fertility” (Ryder 1980:44–45). For Ryder—the pioneer in estimating tempo and quantum—the two were interlinked at both the individual and at the aggregate levels.

Are period quantum and tempo influenced either by different factors or differentially by the same factors? If yes, then they may reflect genuinely distinct processes. If not, they are a single, undifferentiated entity. While instances can be found of changes in timing in reaction to socioeconomic determinants—the Swedish speed premium effect being a very clear-cut case (Andersson 1999; Andersson et al. 2006)—it is not obvious that such instances are exclusively due to timing effects nor that currently available indices of timing would represent them accurately. Research is still needed to address the real-world justification for separating level and timing, at both individual and aggregate levels.

Ultimately, measures used for any purpose need formal validation. No independent criterion is available against which to validate period measures in explanatory studies, but we do have an indirect check on the performance of a period measure as dependent variable: namely, explanatory success. Indicators of period fertility as explanandum would be validated by an empirically successful explanation of period trends in which they were embedded—a form of construct validity. As Ryder has suggested, we will know we have the right measures when we have a good explanation of time trends. Thus far, we lack convincing, well-documented explanations for period trends that could help to adjudicate between different measurement approaches. Nevertheless, success in explanation, rather than a check against cohort values, is the appropriate criterion for evaluating indicators of period fertility as dependent variable.

When data are available on completed childbearing, measuring cohort fertility is straightforward, subject only to the limits of data quality and sample size. However, when cohort childbearing is incomplete, period rates are used in the estimation process, and that presents some challenges. In a paper that has influenced this article, van Imhoff (2001:24–25) remarked: “A particularly important struggle faced by demographic analysts is how to arrive at statements about family formation processes from a cohort perspective . . . from data that are collected on an annual basis. . . .” Synthetic cohort indicators are the principal means by which period fertility is converted into cohort terms. Demographic translation is another approach but will not be discussed here (Keilman 1994; Ryder 1964).

The divergence between time series of period and cohort indicators is one of the best known facts of demography. Nevertheless, the discipline appears prone to a curious doublethink that effectively treats synthetic indices such as the period TFR as cohort estimators. Evidence of the conflation of the two is found in the long-standing criticism that period indices constructed additively occasionally produce results that are impossible in a real cohort (see, e.g., Bongaarts 2002; Bongaarts and Feeney 1998; Ryder 1990; Sobotka 2003; van Imhoff and Keilman 2000). The causes of such apparent anomalies have been known for decades: variable age-specific rates and tempo change. However, that the feature is still mentioned as a defect reveals that the TFR continues to be thought of in some sense as a cohort estimator.

Evidence is mixed on the validity of tempo-adjusted measures as estimators of completed cohort parameters. There are scattered data—proportions having a first birth in the United States or proportions marrying in France—suggesting that adjusted period values can be closer to cohort figures than are the period equivalent, although limited evidence is available thus far (Bongaarts and Feeney 2006: Figures 8–13; Sobotka 2003; Winkler-Dworak and Engelhardt 2004). But time series of annual adjusted TFRs do somewhat less well, not showing a reliable improvement over the conventional TFR in approximating to the total fertility of associated cohorts (Schoen 2004; Smallwood 2002b; Sobotka 2003, 2004a; van Imhoff and Keilman 2000). Thus, evidence validating the effectiveness of tempo adjustment in reducing measurement bias in the period TFR as an indicator of cohort fertility is at best equivocal. Adjustment, in any case, cannot be a complete solution since it cannot correct the bias resulting from the nonconstancy of age-specific rates.

Note that validation against real cohort outcomes is not in keeping with the original rationale advanced by Bongaarts and Feeney (1998) for tempo adjustment (although they also comment that in adjusting for tempo they have “extrapolated the experience of a single year into the future and ascertained the implied cohort fertility” [p. 289]). Nevertheless, such a validity check accords with the thinking of other demographic scholars, for whom the essential purpose of tempo adjustment is to better approximate cohort values (Kohler and Ortega 2002a; Morgan et al. 2009; Schoen 2004; Smallwood 2002b; Sobotka 2003; van Imhoff 2001; van Imhoff and Keilman 2000). Also, in more recent work, Bongaarts and Feeney (2006:145) come closer to this standpoint, seeing the adjusted period measures as “approximate measures of lagged cohorts” when “patterns of change . . . are close . . . to the translation assumptions.”

Period fertility indicators may be used to appraise fertility prospects more generally. A broad reference to the future is found in the discourse of fertility measurement in various ways. It appears to be what van Imhoff (2001:24) had in mind when he said that by “level of fertility,” we mean something like “how many children do people have, on average.” It also seems essentially what is meant by widely used references to “true” or “underlying” fertility, or the completed fertility “implied by” current rates (at least according to one reading—another will be considered in a later section).

On one interpretation, tempo-adjusted measures are a guide to longer-run fertility in some nonspecific sense, although not cohort fertility. This is how tempo adjustment has been viewed in practice by some demographic analysts—as carrying implications for long-run trends in fertility (Bongaarts 2002; Frejka and Ross 2001; Kohler and Ortega 2002a; Lesthaeghe and Willems 1999; Morgan and Berkowitz King 2001). However, evidence validating the TFR_adj as a predictor of longer-run fertility is as yet rather weak, as in the cohort case, and doubts have been expressed as to whether fertility is likely to reach the levels implied by the tempo-adjusted TFR (Frejka and Calot 2001; Frejka and Ross 2001; Lesthaeghe and Willems 1999; Sobotka 2004a).

Another role for period fertility measures is to express population growth prospects, reflected in the convention of regarding a TFR of 2.1 as replacement fertility. The replacement-level TFR idea assumes stable conditions just like its predecessors, the net and gross reproduction rates (NRR and GRR). It has been known since the 1940s that a single period’s TFR is no indication of future growth prospects. And despite the severe criticisms to which reproduction rates have been subject (Dorn 1950; Hajnal 1947, 1959; Stolnitz and Ryder 1949; Whelpton 1946), we continue to use the TFR as a kind of reproduction rate.

It is well known that replacement level fertility is empirically invalid as a concept because the TFR can be below replacement for decades without resulting in population decline (Smallwood and Chamberlain 2005). The reason is, of course, that the long-run, stable assumptions are almost never valid. Hajnal (1959:29) captured the essence of the matter by questioning the sense of “measuring the reproductivity of an actual population that is not stable.” To suggest adjusting the TFR to improve the current estimate of replacement prospects is taking the period estimate of replacement fertility too seriously. Besides, such a proposal goes beyond the evidence. There is no evidence that the TFR_adj improves on the classic TFR as an indicator of long-run population replacement. Validation of the adjusted TFR as an indicator of replacement is lacking hitherto.

When period measures are used to estimate cohort or other future outcomes, there is no argument in principle against tempo adjustment. What matters is the empirical fit between an adjusted indicator and its target. Tempo-adjusted period measures have had limited success in approximating cohort quantities better than unadjusted versions, and the record in predicting future fertility movements is patchy, also. No evidence is at hand on the performance of the TFR_adj as an indicator of replacement.

How does adjustment fare against alternatives? Population science has some well-tried procedures for explicitly gauging likely future trends. One approach is to assess possible futures from detailed analysis of the recent past. Such close analysis has prompted several scholars to question the realism of scenarios implied by adjusted measures (Frejka and Calot 2001; Lesthaeghe and Willems 1999; Sobotka 2004b; van Imhoff 2001). The formal counterpart of this approach—a full population projection—is a further, time-honored alternative to tempo adjustment. Forecasting has several hard-to-beat advantages as a guide either to completed cohort fertility or to future fertility levels. The estimates are presented as projections rather than as measures, their inherent uncertainty is acknowledged, and assumptions about future movements in component rates are made explicit. In anticipating the future, a forecasting methodology will always trump tempo adjustment because it is less constrained, is more flexible, and can incorporate a range of possible forms of timing change, not just the Bongaarts-Feeney variety. Forecasting assumptions can be made selectively and with judgment. They are limited in applied contexts only by their perceived realism. Importantly, forecasts of cohort fertility can use the cumulated fertility of incomplete cohorts at the base period. By contrast, tempo adjustment is a one-trick pony—a rigid application of the same transformation of period rates, year by year—and dispenses with the entire past of the fertility process but for each pair of adjacent periods. To estimate the ultimate mean family size of incomplete cohorts or future period fertility, an explicit forecast is a more transparent, versatile, and powerful vehicle (cf., Kohler and Ortega 2002a, b; Lesthaeghe and Willems 1999; Schoen 2004; van Imhoff 2001). In particular, replacement prospects are gauged more convincingly by a realistic forecast than by any single period’s TFR, adjusted or not.