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Introducing age, period and cohort effects
Andrew Bell
Age, period and cohort (APC) effects are three ways in which societies can change over time, and as such they are of great interest to social scientists across a range of disciplines. However, despite these concepts being fundamental to much social science research, they are poorly understood â in terms of how they can be uncovered, what they really mean and even fundamentally what they are.
This book brings together a collection of perspectives on how applied social scientists should approach age, period and cohort effects. In some cases, this involves complex statistical models; in others, carefully thought through but simple models; in others, data visualization. Why the need for such a plethora of approaches for the apparently simple question of how things change over time? As we will see, the answer is that understanding age, period and cohort effects is not as simple as it may seem at first glance and, as such, attempting to empirically uncover those effects requires making decisions relating to what specifically we want to find out, what assumptions we are able to make and the nature of the data available to us.
In this chapter, I aim to introduce APC effects, both in terms of how they should be understood and the difficulties that modelling them pose. I will do so in relatively simple terms (also see Bell, 2020; Fosse & Winship, 2019 for other accessible introductions to/reviews of the subject). Hopefully, by the end of this introduction, the methodological issues that the subsequent chapters are grappling with will become clear.
What are age, period and cohort effects1
Age effects are perhaps the easiest of the APC trio to understand â as we get older, we become, say, more conservative, or more likely to die, or more religious. There might also be effects that are specific to a particular age â perhaps we become more likely to drink to excess on our 18th/21st birthday, or more likely to buy a sports car around the age of 45.
Period effects are the effect of a particular year â that is the effect of existing in a particular historical moment. The mortality rate of young men is much greater, for instance, during times of war or disease epidemics; mortality rates might also be higher during a recession, as might the likelihood of an individual holding a particular political viewpoint. These are generally associated with discrete events, although we could also imagine long-run, continuous-period effects: for instance, improvements in healthcare or in air quality over time might result in gradual reductions in mortality for all people.
Finally, cohort effects are the effect of being in a particular birth cohort, or generation. Often this is conceived of as the effect of our âformative yearsâ â that is, much of what we think, how healthy we are, and who we are, is defined by the first few years of our lives, and the effect of these early years stays with us throughout the rest of our lives. Again, these could be continuous trends, whereby successive birth cohorts experience better healthcare in their early years, which sets them up to be healthier throughout the rest of their lives. But it could also be a result of discrete events â for instance, wars, pandemics or recessions could, if lived through in our formative years, affect individuals for the rest of their lives. There is strong evidence of such effects on mortality for people born during or just before the Siege of Leningrad or the Spanish Flu pandemic. Those people had higher mortality rates many years after those events took place, because they occurred in their formative years.
In each of these cases, we have seen that APC effects can have both discrete and continuous components; indeed we may have both discrete and continuous effects of one or all of APC. The continuous components explain how things change gradually with one of APC. The discrete components express the effect of being at a particular value of one of APC (on top of any gradual change). This distinction is particularly important throughout this book.
Some readers might already be thinking about some of the conceptual difficulties with understanding and distinguishing between these three. First, all three of APC operate through other variables â that is, it isnât a particular year that has an effect, but the war, or recession, or healthcare policies that are occurring at that time. As such, understanding APC is often only the first step in understanding what is happening. Related to this, many of those other variables could operate through more than one of APC â for instance, a war could have both a period and a cohort effect, as could changes in healthcare policies. It is also possible to imagine interaction effects between each of APC â for instance, a war might have a period effect for only people of a certain age (and gender). Given this, we can see that a simple question (âhow do things change over time?â) is often not simple at all.
Different types of data and identifying APC
There has been a vast increase in the amount of longitudinal data available to researchers, which has made the prospect of empirically uncovering APC effects all the more credible. However, even with cross-sectional data (that is measured at the same time and not longitudinally), it is possible to think through some questions regarding APC. With such data, there is no variation in period, and age and cohort are exactly collinear, so that we cannot know if any differences are the result of cohort differences (when people were born), or age differences (how old people are), although we will often have a good idea based on theory or intuition. Similarly, single cohort studies, that follow a single birth cohort through their lives, have no variation in cohort, and period and age are exactly collinear (although again, we might be more likely to interpret any patterns in a particular way).
However, when analysing APC, we might group multiple cross-sectional studies, or multiple cohorts, together. Alternatively, we might have panel data, which follow the same individuals through time, but measure individuals of all ages on all occasions. In these instances, we have variation in all of APC â however, as we will see in the next section, there remains a difficulty in identifying these effects.
In all these cases, we can see why one of APC might be forgotten about. With cross-sectional data, we might forget about period and cohort, and just consider age. With panel data, we might see a square age-by-year table and think we only need to think about age and year, even though cohort varies in the data as well. Such errors can be problematic, however, and produce a less nuanced, misleading and often incorrect impression of the effects that APC have. However, attempting to consider all three of APC is also problematic, as we will see now.
The identification problem
Age, period and cohort are intrinsically linked, such that the age of any individual is equal to the year of measurement (period), minus their birth year (cohort).
Age = Period â Cohort (1.1)
This is a problem if we want to find the continuous effect of all three of these because, just like two of APC with a single cross section or cohort study, the three variables are exactly collinear. That is not to say that all three couldnât have an effect â indeed in the previous sections we have seen plausible examples of all three of these variables. But it does mean that working out which linear effects are producing the data is often impossible from the data alone.
For instance, consider the following example of a data-generating process that might exist, to explain the changes in peopleâs political opinions:
(1.2)
Here we have a situation where, on average, an individual becomes more right wing as they age; as time goes on (period), people generally become more right wing; and each successive generation (cohort) is more right wing than the last.
Now imagine a different data-generating process
(1.3)
Here, there is no effect of age or cohort, but a stronger effect of period. However, because Age = Period â Cohort, these two data-generating processes would produce exactly the same outcome variable â exactly the same levels of rightwingness. This is a problem if, as a researcher, we are presented with this data, since we cannot know which is true. If we fit a standard regression model, such as
(1.4)
the model would not be able to run, due to the exact collinearity between the three variables in the model.
Instead, we would need to make some kind of assumption, which would push our model to find one equation or the other. The problem is that the difference between these two equations is not a question of a small amount of bias. The difference in how we would interpret these two equations is huge.
Note that, alternatively, we might want to fit the model using dummy variable coding, with a variable for each of the values of age, period or cohort (less a reference category for each). Whilst it is only linear effects that are affected by the identification problem described above, and such an approach would allow non-linear, discrete effects to be modelled, using dummy variables does not solve the problem. Regardless of how we model our data, any linear components of APC effects in the data-generating process will remain in the data. The choice of model would not change the fact that the linear component of those effects would be unable to be told apart, and the model would experience the same problems of exact collinearity between the dummy variables. This is the case even if the data-generating process isnât exactly linear. Different chapters in this book refer to models that use both linear and dummy effects of APC, but in each case, the underlying effects in the data will often be a mixture of linear, continuous effects and non-linear effects. Whilst the latter can be identified, the former cannot, unless we are willing to make some quite strong assumptions about APC.
That is the key point: we cannot identify linear trends in APC without making some quite strong assumptions about at least one of APC, and whilst we can identify non-linear patterns around those trends, they may be difficult to understand without the linear trends around which they vary. The assumptions and approaches that we choose to help us to understand these patterns will have a big effect on the results that we find. The next section outlines some of those approaches, including those demonstrated in the rest of this book.
What we should and shouldnât do: the chapt...
