## Abstract

### Objective

Identify the effect of differences in criminal activity among birth cohorts on crime rates over time. Determine the extent to which cohort effects are responsible for nationwide crime reductions of the last thirty years.

### Methods

Use a panel of state age-arrest data and frequently used economic, social, and criminal justice system covariates to estimate a proxy or characteristic function for current period effects. Combine these results with national age-arrest data to estimate nationwide age, current period, and birth cohort effects on crime rates for 1980–2016.

### Results

Criminal activity steadily declined between the 1916 and 1945 birth cohorts. It increased among Baby Boomers and Generation X, then dropped rapidly among Millennials, born after 1985. The pattern was similar for all index crimes. Period effects were mostly responsible for the late 1980s crack boom and the 1990s crime drop, but age and cohort effects were primarily responsible for crime rate reductions after 2000. In general, birth cohort and current period effects are about equally important in determining crime rates.

### Conclusions

Policies aimed at reducing delinquency among young children may be more effective in the long run than current policies aimed at incapacitation, deterrence, and opportunity reduction.

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## Notes

- 1.
In the same vein, O’Brien (2019) supplemented his (fundamentally cohort-proxy) analysis of U.S. murder rates by assuming a unimodal age curve. These assumptions are perfectly sensible; as shown below, they are also consistent with the results presented in the text. In the absence of an alternative solution, however, they would remain unverifiable. This may be less of a problem for other dependent variables or in other fields of study.

- 2.
- 3.
Arrest counts were assigned to one-year bins for all age groups where BJS provides data in two- to five-year bins so as to minimize the sum of squared annual rate changes, subject to the constraint that the sum of all arrest counts remain the same as the corresponding bin in the original data. This produces almost identical results to the method of differences (Steffensmeier et al. 1992) or B-splines (McNeil et al. 1977), but preserves the sum of arrest counts. The lack of anomalies in the empirical data suggests that the minimization function is reasonable. Further details are available from the author.

- 4.
Sampling decisions were guided by a “90/90 rule”: If 90% of the population were covered by law enforcement agencies reporting age-arrest statistics in 90% of years (that is, 39 of 43 years), then it could be included in the sample. A few adjustments were made in the sampling process. Delaware was not included because 27% of all arrests were attributed to the Delaware State Police, which does not cover a unique population and did not report reliably. Hawaii was not included because it was unlikely to be representative of the rest of the country. Maine was chosen instead of Rhode Island because 90% of its population was always covered. Iowa was added to improve geographic coverage. Wyoming was combined with Idaho. Thus panel size

*N*= 15. - 5.
Feasible generalized least squares is more accurate than ordinary least squares in this context, but the standard errors are systematically too small. The bootstrap is necessary to cure this problem (Moundigbaye et al. 2018). An earlier version combining OLS with Driscoll-Kraay standard errors (which are robust to heteroskedasticity, spatial dependence, and serial correlation) produced very similar coefficients, trends, and standard errors.

- 6.
Differences in twice-lagged prisons and in state government spending were positively correlated with murder, rape, and assault differences. Gubernatorial elections were valid for all crimes, but were too weak a predictor of prison rates to be useful.

- 7.
To reduce the problem of capitalizing on chance, Bayesian model averaging was used in the first stage of the two-stage instrumental variables process. For all crime types, first-stage

*F*> 10.0 with average*F*= 13.323; average overidentification*F*= 1.353 and*p*(*F*) > .10 for all crime types. In addition to demonstrating the validity of the instruments, the overidentification test also shows directly that the covariates are not significantly associated with age or cohort effects. Per Winship and Harding (2008), we could enter these covariates into an APC model in place of period effects and expect to get valid results. - 8.
It may be possible to avoid the omitted variable analysis if the covariates in Eq. (8) account for enough of the variance in Δ

*p*_{t}that the remainder can be chalked up to random errors. One measure of this is to compare*R*^{2}in Eq. (1) to R^{2}when the Δ*p*_{t}are replaced by their predicted values from Eq. (8). In this case the difference is significant for all crime types, indicating that some covariates have in fact been omitted. My thanks to an anonymous reviewer for the suggestion. - 9.
Specifically, the average values of κ

_{2}for seven index crimes over the 1980–2015 time period for Australia, Canada, Denmark, Finland, Germany, Iceland, New Zealand, Norway, Sweden, and Switzerland are highly correlated with one another (average inter-country*r*= + .831, Cronbach’s α = + .980). The mean κ_{2}of the ten countries is correlated with U.S. mean α over the same time period with*r*= + .896. - 10.
Age effects are often controlled for by including the proportion of the population in crime-prone ages (usually 18–24) as a covariate. The importance of age effects suggests that more such covariates – and perhaps more specific ones – should be used as controls.

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## Acknowledgements

My thanks to Jonathan Davis for suggesting the omitted variable analysis, and to two anonymous reviewers for a wide variety of insightful suggestions. This paper is much improved due to their assistance. Research support was provided by the Policy Research Institute, University of Texas at Austin.

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## Appendix. Derivation of Level Estimates of π

### Appendix. Derivation of Level Estimates of π

Estimates of π are relatively straightforward for regression in differences. The intercept is the expected change in *P* if there is no change in the *X* variables, almost a definition of a secular trend. For the age-arrest data set, there is considerable serial correlation in age, period, and cohort deviates, more or less requiring use of a differenced model. This will not always be true, however, so it is worth considering how to estimate π from a levels regression.

Let us assume that period effect *P*_{t} is a function of *j* causal variables *X*_{jt}, such that

*P*_{t} is presumed to be stationary about a deterministic trend, such that *P*_{t} = π*t* + *p*_{t}. Predictor variables *X*_{jt} are likewise stationary about deterministic trends, *X*_{jt} = δ_{j}* t* + *x*_{jt}. *P*_{t} and π are unknown, but we may reasonably assume that π is a weighted average of the predictor trends,

Equation (A.1) cannot be estimated directly, but we can use Eq. (A.2) to estimate π if the β_{j} can be obtained through some alternative procedure. Since *p*_{t} is identified and Eq. (A-1) can be rearranged as

we also may be able to save a step and obtain standard errors directly.

A complete proof involves *j* + 1 variables and requires matrix algebra, but it is not difficult to prove out Eq. (A.3) for the case of one predictor. We can generalize from there.

### One Causal Predictor

Assume *p*_{t} depends only on one causal variable. Then Eq. (A-3) may be estimated as

Let us solve first for *b*, then for *c*.

*Solve for b.* As usual for regressions with two independent variables, the least squares estimate of coefficient *b* can be obtained from

where, for example, Σ*Xt* refers to the sum from 1 to *T* of (\(X\) _{t} – \(\stackrel{-}{X}\))(*t* – \(\stackrel{-}{t}\)). Since *p* has been detrended, *p* and *t* are asymptotically uncorrelated. Thus Σ*tp* → 0 and the second term in the numerator drops out. This leaves

Breaking *X* into its component parts,

Since *x* and *t* are asymptotically uncorrelated, Σ*xt* → 0, so this simplifies to

Note that this is the formula for the coefficient on *x* in the simple regression, *p*_{t} = *a* + *bx*_{t} + *e*_{t}. Adding the trend term allows for consistent estimates of the effect of *x*, even if both variables are trending and only one has been detrended. Thus plim *b* = β and (in the two-variable case) π = *b*δ.

*Solve for c.* The least squares estimate of coefficient *c* can be obtained from

As before, Σ*tp* → 0, so the first term in the numerator drops out. This leaves

Again breaking *X* into its component parts,

Since Σ*xt* → 0, this simplifies to

\(\begin{aligned} {\text{Because}}\;{\text{plim }}b & = & \, \Sigma xp/\Sigma x^{2} , \\ {\text{plim }}c & = & \, {-}b\delta \\ & = & \, {-}{\uppi }. \\ \end{aligned}\).

The coefficient on *t* is a consistent estimator of the trend in *P*_{t}.

### Multiple Causal Predictors

If we have multiple causal variables, we estimate Eq. (A-3). This may be reframed as two-variable regression,

where, per Eq. (A-1),

If we set θ = Σ_{j} β_{j} δ_{j}, a weighted average of the causal variable trends, and *z*_{t} = Σ_{j} β_{j}* x*_{jt}, a weighted average of the detrended causal variables, then *Z*_{t} = θ*t* + *z*_{t}. From Eq. (A.7), *b* = 1 and (from the two-variable case) *c* = –*b*θ = –π*.*

Previous studies have shown that estimates *b* and *c* are Normal-distributed and converge at nominal rates (García-Belmonte and Ventosa-Santaulària, 2011), so standard errors and confidence intervals can be computed and interpreted in the usual way.

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Spelman, W. How Cohorts Changed Crime Rates, 1980–2016.
*J Quant Criminol* (2021). https://doi.org/10.1007/s10940-021-09508-7

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### Keywords

- Age-period-cohort models
- Crime
- Model identification
- Birth cohorts
- Primary prevention