What is the Average Annual Crime Rate in the United States?

David McDowallSchool of Criminal Justice

University at Albany, SUNY135 Western AvenueAlbany, NY 12222

mcdowall@albany.edu

Is an “average annual crime rate” a sensible concept?

If such a rate existed, notions like rates that were “too high”or “too low” would be meaningful.

It would be possible to predict increases and decreases incrime with high confidence.

The presentation will mostly argue against this possibility.

Is an “average annual crime rate” a sensible concept?

If such a rate existed, notions like rates that were “too high”or “too low” would be meaningful.

It would be possible to predict increases and decreases incrime with high confidence.

The presentation will mostly argue against this possibility.

Stylized facts: empirical characteristics typical across diversesets of data.

Are useful for developing and testing theory.

The presentation will consider stylized facts about the meanor level of the series.

The focus will be on the link between theoretical andempirical issues.

Stylized facts: empirical characteristics typical across diversesets of data.

Are useful for developing and testing theory.

The presentation will consider stylized facts about the meanor level of the series.

The focus will be on the link between theoretical andempirical issues.

Rates of Murder, Rape, Robbery, Assault, Burglary, and Motor Vehicle Theft, 1933-2009

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Murders per 100,000

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Rapes per 100,000

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Burglaries per 100,000

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Assaults per 100,000

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Robberies per 100,000

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Motor vehicle thefts per 100,000

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The means in the previous slide describe the series.

But do they reflect the underlying process that generatedthem?

Did the means hold in the past, and will they continue tohold in the future?

Certainly, the variation in the series has little affinity for themeans.

The means in the previous slide describe the series.

But do they reflect the underlying process that generatedthem?

Did the means hold in the past, and will they continue tohold in the future?

Certainly, the variation in the series has little affinity for themeans.

Theories that imply a constant mean:

(1). Functionalist (Durkheim, Erikson).

Definitions of crime vary to keep levels of crime atconstant value.

Theories that imply a constant mean:

(1). Functionalist (Durkheim, Erikson).

Definitions of crime vary to keep levels of crime atconstant value.

(2). Economic.

Natural rate of crime exists, like natural rate ofunemployment.

As crime rises, citizens demand more protection. When it reaches natural rate, they lose interest.

Crime then fluctuates around the natural rateequilibrium.

(3). Institutional anomie (Messner and Rosenfeld).

(2). Economic.

Natural rate of crime exists, like natural rate ofunemployment.

As crime rises, citizens demand more protection. When it reaches natural rate, they lose interest.

Crime then fluctuates around the natural rateequilibrium.

(3). Institutional anomie (Messner and Rosenfeld).

(3). Institutional anomie (Messner and Rosenfeld).

Proposes a natural rate of crime, constant withinnations but variable between nations.

Rate depends on specific institutional configurationwithin a nation.

In principle, a nation’s natural rate of crime might bepredicted from theoretical considerations.

(3). Institutional anomie (Messner and Rosenfeld).

Proposes a natural rate of crime, constant withinnations but variable between nations.

Rate depends on specific institutional configurationwithin a nation.

In principle, a nation’s natural rate of crime might bepredicted from theoretical considerations.

Many alternatives to constant mean processes exist.

Perhaps the most obvious, however, is a process in whichthe mean changes from period to period.

Differently stated, such a process lacks a mean.

This is an integrated (nonstationary) process.

Two major types of integrated processes are trends andrandom walks.

In both cases, the process mean varies over time and theprocess variance increases.

Many alternatives to constant mean processes exist.

Perhaps the most obvious, however, is a process in whichthe mean changes from period to period.

Differently stated, such a process lacks a mean.

This is an integrated (nonstationary) process.

Two major types of integrated processes are trends andrandom walks.

In both cases, the process mean varies over time and theprocess variance increases.

Stationary series

Mean Standard deviation observations observations 1-100 1.96 1-100 1.09 1-200 1.96 1-200 1.06 1-300 1.96 1-300 1.02 1-400 2.01 1-400 1.02 1-500 2.00 1-500 1.03

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Deterministic trend

Mean Standard deviation observations observations 1-100 2.46 1-100 2.45 1-200 4.99 1-200 4.99 1-300 7.51 1-300 7.51 1-400 10.00 1-400 10.00 1-500 12.42 1-500 12.52

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Stochastic trend

Mean Standard deviation observations observations 1-100 -6.82 1-100 4.60 1-200 -4.16 1-200 4.84 1-300 -3.00 1-300 4.69 1-400 -3.41 1-400 4.27 1-500 -2.14 1-500 5.04

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Random walk

Mean Standard deviation observations observations 1-100 1.48 1-100 3.37 1-200 1.61 1-200 3.22 1-300 .93 1-300 3.17 1-400 -1.18 1-400 5.49 1-500 -1.20 1-500 5.94

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Equations for time series processes

Stationary process:

Yt = b0 + at

Random walk:

Yt = Yt-1+ at

Deterministic trend:

Yt = b0 + b1(timet) + at

Stochastic trend:

Yt = b0 + Yt-1+ at

The variables driving a random walk might be predictable individually, but their collective impact is not predictable. Have been useful models of financial behavior (e.g., stock market prices). Are generally consistent with existing explanations of the crime drop that began in the 1990’s.

Random walks do not possess means or meaningful long-term trends. Relationships are (in most cases) only short-term. Analysis must (in most cases) be conducted in differences:

Yt = (Yt - Yt-1) + at

Random walk

Random walk, differenced

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Results of Tests for Nonstationarity (Unit Roots)

United States, 1933 - 2009 Null hypothesis: Crime ratesfollow nonstationary process

Murder Fail to rejectRape Fail to rejectRobbery Fail to rejectAssault Fail to rejectBurglary Fail to rejectAuto Theft Fail to reject

Results of Tests for Nonstationarity (Unit Roots)

United States, 1933 - 2009 Null hypothesis: Crime ratesfollow nonstationary process

Murder Fail to rejectRape Fail to rejectRobbery Fail to rejectAssault Fail to rejectBurglary Fail to rejectAuto Theft Fail to reject

Results of Tests for Nonstationarity (Unit Roots), Continued

U.S. cities with populations exceeding 400,000, 1960 - 2009Null hypothesis: Murder rate follows nonstationary process

Fail to reject:

New York San Francisco El Paso Long BeachLos Angeles Jacksonville Seattle AlbuquerqueChicago Columbus Nashville Kansas CityHouston Austin Fort Worth AtlantaPhiladelphia Baltimore PortlandPhoenix Memphis Oklahoma CitySan Diego Milwaukee New OrleansDetroit Boston Las VegasSan Jose Washington Cleveland

Reject:

Indianapolis Denver Charlotte Tucson Sacramento

Results of Tests for Nonstationarity (Unit Roots), Continued

U.S. cities with populations exceeding 400,000, 1960 - 2009Null hypothesis: Murder rate follows nonstationary process

Fail to reject:

New York San Francisco El Paso Long BeachLos Angeles Jacksonville Seattle AlbuquerqueChicago Columbus Nashville Kansas CityHouston Austin Fort Worth AtlantaPhiladelphia Baltimore PortlandPhoenix Memphis Oklahoma CitySan Diego Milwaukee New OrleansDetroit Boston Las VegasSan Jose Washington Cleveland

Reject:

Indianapolis Denver Charlotte Tucson Sacramento

But crime rates do not follow simple random walkprocesses.

If they are random walk nonstationary, they also containadditional structure that smooths them.

U.S. Homicide Rate Simulated Random Walk

But crime rates do not follow simple random walkprocesses.

If they are random walk nonstationary, they also containadditional structure that smooths them.

U.S. Homicide Rate Simulated Random Walk

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Autoregressive models for the differences can account fortheir smooth appearance.

First-order autoregressive model for the differences:

Yt = b0 + b1(Yt-1) + et

Second-order autoregressive model for the differences:

Yt = b0 + b1(Yt-1) + b2(Yt-2) + et

Autoregressive models imply that the effects of externalshocks persist, and that they diffuse over time.

They also imply that crime rates should be highlypredictable over short periods.

Autoregressive models for the differences can account fortheir smooth appearance.

First-order autoregressive model for the differences:

Yt = b0 + b1(Yt-1) + et

Second-order autoregressive model for the differences:

Yt = b0 + b1(Yt-1) + b2(Yt-2) + et

Autoregressive models imply that the effects of externalshocks persist, and that they diffuse over time.

They also imply that crime rates should be highlypredictable over short periods.

Homicide Rate, 1925-2000Model: ΔYt = b0 + b1(ΔYt-1) + et

Parameter Estimate Standard Error t

b0 -.0061 .0121 -.51b1 .4567 .1048 4.36_____________________________________________________

Rape Rate, 1933-2001Model: ΔYt = b0 + b1(ΔYt-1) + et

Parameter Estimate Standard Error t

b0 .0298 .0110 2.72b1 .2323 .1205 1.93____________________________________________________

Robbery Rate, 1933-2001Model: ΔYt = b0 + b1(ΔYt-1) + et

Parameter Estimate Standard Error t

b0 .0116 .0199 .56b1 .4708 .1034 4.55_____________________________________________________

Assault Rate, 1933-2001Model: ΔYt = b0 + b1(ΔYt-1) + b2(ΔYt-2) + et

Parameter Estimate Standard Error t

b0 .0286 .0130 2.20b1 .3016 .1188 2.54b2 .2934 .1188 2.47_____________________________________________________

Burglary Rate, 1933-2001Model: ΔYt = b0 + b1(ΔYt-1) + et

Parameter Estimate Standard Error t

b0 .0147 .0149 .99b1 .4827 .1046 4.61_____________________________________________________

Motor Vehicle Theft Rate, 1933-2001

Model: ΔYt = b0 + b1(ΔYt-1) + et

Parameter Estimate Standard Error t

b0 .0122 .0183 .66b1 .5086 .1047 4.86

Although the autoregressive models fit the series, theyrequire more complicated theoretical explanations.

And this raises the possibility that simpler alternativemodels may be preferable.

Although the autoregressive models fit the series, theyrequire more complicated theoretical explanations.

And this raises the possibility that simpler alternativemodels may be preferable.

A nonlinear process is one possibility.

A nonlinear process operates differently depending on thecurrent value of a series.

Threshold Autoregressive (TAR) models are an example. One (of many) possible TAR models is:

ΔYt = b0 + b1(Yt-1) + et if Yt-1 c

ΔYt = b2 + b1(Yt-1) + et if Yt-1 > c

Here the series follows different processes depending onwhether the previous level was above or below a threshold.

A series may therefore switch between different equilibriadepending on conditions.

A nonlinear process is one possibility.

A nonlinear process operates differently depending on thecurrent value of a series.

Threshold Autoregressive (TAR) models are an example. One (of many) possible TAR models is:

ΔYt = b0 + b1(Yt-1) + et if Yt-1 c

ΔYt = b2 + b1(Yt-1) + et if Yt-1 > c

Here the series follows different processes depending onwhether the previous level was above or below a threshold.

A series may therefore switch between different equilibriadepending on conditions.

Social interaction and contagion models provide theoreticalsupport for nonlinear processes.

Crimes may be easier to commit when many people arecriminals, and harder to commit when criminals are rare.

Contagion models emphasize the fact that some outcomesgrow explosively once they reach a tipping point.

Social interaction and contagion models provide theoreticalsupport for nonlinear processes.

Crimes may be easier to commit when many people arecriminals, and harder to commit when criminals are rare.

Contagion models emphasize the fact that some outcomesgrow explosively once they reach a tipping point.

Most existing work on nonlinearity begins from a model ofhow a series would vary if it were nonlinear.

It then applies the model to data, and rejects it if it providesa bad fit.

Some approaches simply assume that nonlinearity must exist.

M.B. Gordon, A Random Walk in the Literature onCriminality: A Partial and Critical Review on SomeStatistical Analyses and Modelling Approaches. European Journal of Applied Mathematics, 2010, 21:283-306.

This is not a good testing strategy.

Most existing work on nonlinearity begins from a model ofhow a series would vary if it were nonlinear.

It then applies the model to data, and rejects it if it providesa bad fit.

Some approaches simply assume that nonlinearity must exist.

M.B. Gordon, A Random Walk in the Literature onCriminality: A Partial and Critical Review on SomeStatistical Analyses and Modelling Approaches. European Journal of Applied Mathematics, 2010, 21:283-306.

This is not a good testing strategy.

A different approach is to allow a linear model to explain aseries, and then test for remaining nonlinear features.

Such tests are extremely conservative, and they have beenrarely applied.

However, they do not provide support for nonlinearalternatives.

McDowall, “Tests of Nonlinear Dynamics in U.S.Homicide Time Series, and Their Implications.” Criminology, 2002.

McDowall and Loftin, “Are U.S. Crime Rate TrendsHistorically Contingent?” Journal of Research in Crime andDelinquency, 2005.

A different approach is to allow a linear model to explain aseries, and then test for remaining nonlinear features.

Such tests are extremely conservative, and they have beenrarely applied.

However, they do not provide support for nonlinearalternatives.

McDowall, “Tests of Nonlinear Dynamics in U.S.Homicide Time Series, and Their Implications.” Criminology, 2002.

McDowall and Loftin, “Are U.S. Crime Rate TrendsHistorically Contingent?” Journal of Research in Crime andDelinquency, 2005.

A second possibility is that structural breaks change thecrime-generating process.

For example:

ΔYt = b0 + b1 Δ(Yt-1) + et if t m

ΔYt = b2 + b1 Δ(Yt-1) + et if t > m

Here the effects of external variables operate differently atsome periods than at others.

This implies very strong theories: the causes of crimechange abruptly and fundamentally at specific time points.

The alternative is perhaps more attractive to criminologists: the causes of crime are basically the same now as even onehundred years ago.

A second possibility is that structural breaks change thecrime-generating process.

For example:

ΔYt = b0 + b1 Δ(Yt-1) + et if t m

ΔYt = b2 + b1 Δ(Yt-1) + et if t > m

Here the effects of external variables operate differently atsome periods than at others.

This implies very strong theories: the causes of crimechange abruptly and fundamentally at specific time points.

The alternative is perhaps more attractive to criminologists: the causes of crime are basically the same now as even onehundred years ago.

Some recent work claims evidence for two breaks and threetrends since 1960.

Jemma Cook and Steve Cook, Are US Crime RatesReally Unit Root Processes? Journal of QuantitativeCriminology, forthcoming.

Considers only the period since 1960, however.

Other breaks would have to exist over the period since1933, making the series little different from a random walk.

The work supporting the two break finding also requireslinear trends during each of the periods.

These seem highly unlikely.

Some recent work claims evidence for two breaks and threetrends since 1960.

Jemma Cook and Steve Cook, Are US Crime RatesReally Unit Root Processes? Journal of QuantitativeCriminology, forthcoming.

Considers only the period since 1960, however.

Other breaks would have to exist over the period since1933, making the series little different from a random walk.

The work supporting the two break finding also requireslinear trends during each of the periods.

These seem highly unlikely.

Overall, although it is not airtight, the evidence favorslinear, autoregressive random walks.

This is consistent with both criminological theory andempirical analysis.

The autoregressive random walk explanation is broad and insome ways little more than common sense.

Additional theoretical development would be useful.

It is also not very exciting.

But perhaps the unexciting nature of crime rate trends is thekey to understanding them.

Overall, although it is not airtight, the evidence favorslinear, autoregressive random walks.

This is consistent with both criminological theory andempirical analysis.

The autoregressive random walk explanation is broad and insome ways little more than common sense.

Additional theoretical development would be useful.

It is also not very exciting.

But perhaps the unexciting nature of crime rate trends is thekey to understanding them.

More generally, stylized facts about crime provide a way todevelop theories and a constraint on the form that theoriescan take.

Stylized facts do not exist independently of a theoreticalunderstanding of how crime rates might vary.

They are abstractions, and detecting them requiresknowledge of what to look for and of where to look.

Still, it seems reasonable that one might learn somethingabout crime rate trends through an examination of theirfeatures.

More generally, stylized facts about crime provide a way todevelop theories and a constraint on the form that theoriescan take.

Stylized facts do not exist independently of a theoreticalunderstanding of how crime rates might vary.

They are abstractions, and detecting them requiresknowledge of what to look for and of where to look.

Still, it seems reasonable that one might learn somethingabout crime rate trends through an examination of theirfeatures.