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Setting bounds on age, period, and cohort effects using observed data

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Abstract

This paper presents a method that uses observed data from an age-period table to set bounds on the age, period, and cohort effects in an age-period-cohort multiple classification (APCMC) model. The rationale is that with enough periods over a long time span the age distributions within periods on the dependent variable will be affected by different sets of cohorts for each of the periods. This is likely to result in different trends in these separate period age distributions such that the trends in the age distributions will encompass the trend in the age effects that generated the dependent variable values. This approach can help to identify bounds that likely encompass the age, period, cohort parameters that generated the data. The data used in this papers are estimated homicide arrests by single years for those aged 15–64 for the periods 1964 to 2019 in the United States. I utilize the observed trends in the age-distributions for each of the 56 periods as different constraints on the trends for the age effects in the APCMC fixed effects model. These estimates are used to form bounds on the age effects, period effects, and cohort effects.

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Notes

  1. In the fixed effect APCMC model an infinity of solutions fit the data equally well. Most researchers are interested in which solution comes closest to the parameters that actually generated the dependent variable values. I label these parameters as the data generating parameters.

  2. See O’Brien (2016) for a critique of this assumption. The question about this assumption is: Why should a linear trend without much variability around the trend (usually considered a good fit to the trend of the data and that makes the confidence limit for the trend smaller) be considered an argument for a zero linear trend? Or on the other hand: Is a poor linear fit, a good argument for confidence in a linear trend?

  3. As Glenn (2005) notes: “... a definitive separation of age, period, and cohort effects is not just difficult, but impossible. However,... a definitive separation of the effects is not necessary in order for cohort analysis to be useful.”.

  4. Using arrest rates by age or estimated homicide rates by age would not affect the pattern of results from the APCMC model, and if the reader prefers, they can think of the pattern of results as applying to homicide arrest rates.

  5. I used the constrained regression package in Stata (Stata Corp 2019) to conduct the constrained regression.

  6. This was accomplished by constraining the age effect in the APCMC model to one of the values for the age effects using the minimum trend and then one of the age effects using the maximum trend for the age observations within in periods in Step 4 of the Computing Bounds section above.

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  1. Department of Sociology, University of Oregon, Eugene, OR, USA

    Robert M. O’Brien

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O’Brien, R.M. Setting bounds on age, period, and cohort effects using observed data. Qual Quant 57, 2841–2857 (2023). https://doi.org/10.1007/s11135-022-01503-9

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