How Cohorts Changed Crime Rates, 1980–2016

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.

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

$$P_{t} = \, \alpha \, + \, \Sigma_{j} \beta_{j} X_{jt} + \varepsilon_{t} .$$

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P_t is presumed to be stationary about a deterministic trend, such that P_t = πt + 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,

$${\uppi } = \, \Sigma_{j} \beta_{j} \delta_{j} .$$

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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

$$p_{t} = \, \alpha \, + \, \Sigma_{j} \beta_{j} X_{jt} {-}{\uppi }t + \, \varepsilon_{t} ,$$

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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

$$p_{t} = a + b \, X_{t} + ct + e_{t} .$$

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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

$$b = \, \left[ {\Sigma t^{2} \Sigma Xp{-} \, \Sigma Xt\;\Sigma tp} \right] \, / \, \left[ {\Sigma X^{2} \Sigma t^{2} {-} \, \left( {\Sigma Xt} \right)^{2} } \right]$$

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where, for example, ΣXt 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 Σtp → 0 and the second term in the numerator drops out. This leaves

$${\text{plim}}\;b = \Sigma t^{2} \Sigma Xp/\left[ {\Sigma X^{2} \Sigma t^{2} {-} \, \left( {\Sigma Xt} \right)^{2} } \right].$$

Breaking X into its component parts,

$$\begin{aligned} {\text{plim}}\;b & = & \, \Sigma t^{2} \Sigma \, \left( {\delta t + x} \right)p/\left[ {\Sigma \, \left( {\delta t + x} \right)^{2} \Sigma t^{2} {-} \, \left( {\Sigma \, \left( {\delta t + x} \right)t} \right)^{2} } \right] \\ & = & \, \Sigma t^{2} \Sigma \, \left( {\delta tp + xp} \right)/\left[ {\Sigma \, \left( {\delta^{2} t^{2} + \, 2\delta xt + x^{2} } \right) \, \Sigma t^{2} {-} \, \left( {\Sigma \, \left( {\delta t^{2} + xt} \right)} \right)^{2} } \right] \\ & = & \, \delta \, \Sigma t^{2} \Sigma tp + \, \Sigma t^{2} \Sigma xp/\left[ {\delta^{2} \Sigma t^{2} \Sigma t^{2} + \, 2\delta \, \Sigma xt\Sigma t^{2} + \, \Sigma x^{2} \Sigma t^{2} {-} \, \left( {\delta \, \Sigma t^{2} + \, \Sigma xt} \right)^{2} } \right]. \\ \end{aligned}$$

Since x and t are asymptotically uncorrelated, Σxt → 0, so this simplifies to

$$\begin{aligned} {\text{plim}}\;b & = & \, \Sigma t^{2} \Sigma xp/\left[ {\delta^{2} \left( {\Sigma t^{2} } \right)^{2} + \, \Sigma x^{2} \Sigma t^{2} {-} \, \delta^{2} \left( {\Sigma t^{2} } \right)^{2} } \right] \\ & = & \, \Sigma t^{2} \Sigma xp/ \, \Sigma x^{2} \Sigma t^{2} \\ & = & \, \Sigma xp/ \, \Sigma x^{2} . \\ \end{aligned}$$

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 = bδ.

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

$$c = \, \left[ {\Sigma X^{2} \Sigma tp{-} \, \Sigma Xt\Sigma Xp} \right]/\left[ {\Sigma X^{2} \Sigma t^{2} {-} \, \left( {\Sigma Xt} \right)^{2} } \right].$$

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

$${\text{plim }}c = \, {-} \, \Sigma Xt\Sigma Xp/\left[ {\Sigma X^{2} \Sigma t^{2} {-} \, \left( {\Sigma Xt} \right)^{2} } \right].$$

Again breaking X into its component parts,

$$\begin{aligned} {\text{plim}}\;c & = & \, {-} \, \Sigma \, \left( {\delta t + x} \right)t\Sigma \left( {\delta t + x} \right)p/\left[ {\Sigma \, \left( {\delta t + x} \right)^{2} \Sigma t^{2} {-} \, \left( {\Sigma \, \left( {\delta t + x} \right)t} \right)^{2} } \right] \\ & = & \, {-} \, \Sigma \, \left( {\delta t^{2} + xt} \right) \, \Sigma \left( {\delta tp + xp} \right)/\left[ {\Sigma \, \left( {\delta^{2} t^{2} + \, 2\delta xt + x^{2} } \right) \, \Sigma t^{2} {-} \, \left( {\Sigma \, \left( {\delta t^{2} + xt} \right)} \right)^{2} } \right] \\ & = & \, {-} \, \Sigma \, \left( {\delta t^{2} + xt} \right) \, \Sigma \left( {\delta tp + xp} \right)/\left[ {\Sigma \delta^{2} t^{2} \Sigma t^{2} + \, 2\delta \, \Sigma xt\Sigma t^{2} + \, \Sigma x^{2} \Sigma t^{2} {-} \, \left( {\delta \, \Sigma t^{2} + \, \Sigma xt} \right)} \right)^{2} ]. \\ \end{aligned}$$

Since Σxt → 0, this simplifies to

$$\begin{aligned} {\text{plim }}c & = & \, {-}\delta \, \Sigma t^{2} \Sigma xp/\left[ {\delta^{2} \left( {\Sigma t^{2} } \right)^{2} + \, \Sigma x^{2} \Sigma t^{2} {-} \, \delta^{2} \left( {\Sigma t^{2} } \right)^{2} } \right] \\ & = & \, {-}\delta \, \Sigma xp/ \, \Sigma x^{2} . \\ \end{aligned}$$

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

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,

$$p_{t} = a + \, 1Z_{t} + ct + e_{t}$$

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where, per Eq. (A-1),

$$\begin{aligned} Z_{t} = & \, \Sigma_{j} \beta_{j} X_{jt} \\ = & \, \Sigma_{j} \beta_{j} \left( {\delta_{i} t + x_{jt} } \right) \\ = & \, \Sigma_{j} \beta_{j} \delta_{j} t + \, \Sigma_{j} \beta_{j} x_{jt} \\ \end{aligned}$$

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 = θt + z_t. From Eq. (A.7), b = 1 and (from the two-variable case) c = –bθ = –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.