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Friday, December 29, 2006

Is cop killing an inferior activity?

Economists define an inferior good or activity as one whose demand or prevalence increases as income declines, and vice-versa. Thus, the question asked here is whether killing police officers is something that decreases with income. It sounds like a reasonable hypothesis. After all, those with higher income levels have less (negative) interaction with the police, tend to have firearms for hunting and self-protection rather than asset accumulation, and face greater costs, both direct and indirect, from engaging in gunfire with the police. Therefore, economic intuition would suggest that as income increases, ceteris paribus, the number of police killed should decline.

How would you go about testing whether killing police is inferior? By definition, an inferior good or activity is one which declines with an increase in income. Perhaps one way to test this would be to measure the income of those who do and do not engage in killing police. The problem with this approach would be the vast numbers of people who do not actually engage in killing police officers (even if they secretly want to) and this would make it nearly impossible to determine whether, in general, killing police declines with income. However, perhaps we could use aggregate macro-level data over time and utilize the variation in killed policemen, GDP, and population over time.

What made me think about this question was this story which reports that 150 officers were killed in the line of duty in 2006 (73 of them in traffic accidents). While visiting the National nLlaw Enforcement Officers Memorial Fund's website I came across a page that reports the number of officers killed each year in the United States from 1792 through 2005.

Here is the beginning of an idea. I have annual police deaths (some years are not reported and other years report numbers that seem difficult to believe), now I just have to match these data with national income and other information.

I grabbed the data and threw them into Excel so that I could re-organize it. I went over to eh.net and grabbed their information on national population, nominal and real income. I imported that into STATA. Total time, at that point, was about five minutes.

I created a time-series plot of the annual officers killed:

The red lines are 1929 and 1941, respectively. I find the trends after 1900 intriguing. Therefore, I created a few additional variables: dummy variables for prohibition, the war on drugs (starting in 1980), civil unrest (1964-1973), whether we were a nation at war, and whether the national economy was in a recession. I determined the civil unrest variable by looking at the list of race riots in the United States over at wikipedia.

Here are the descriptive statistics of the sample I use - 1900 through 2005:

Variable | Obs Mean Std. Dev. Min Max
lnkilled | 106 4.925818 .4251579 3.496508 5.602119
lngdppc | 106 9.412029 .6723684 8.505728 10.52492
lnpop | 106 11.97215 .3902354 11.23973 12.60064
war | 106 .2264151 .420499 0 1
recession | 106 .1226415 .3295836 0 1
prohibition | 106 .1415094 .3502021 0 1
warondrugs | 106 .1603774 .3686989 0 1
civilunrest | 106 .0943396 .2936892 0 1

There is likely to be autocorrelation in the data, which I account for by including a once-lagged value of officers killed (the twice lagged value was not significant and therefore is not included here).

I regressed the log of officers killed against the log of per-capita income, the log of population, the once lagged value of the dependent variable and the indicator variables. The parameter estimates, if we believe them, can be interpreted as elasticities. The original hypothesis was that killing police is an inferior activity, in other words as income increases, ceteris paribus, we should see fewer cops killed or a negative income elasticity.

Here are the results using data from 1900 through 2006 (with a little over ten minutes spent on the actual analysis):

. reg lnkilled lngdppc lnpop war recession prohibition warondrugs civilunrest l.lnkilled if year
> >1899,r

Regression with robust standard errors Number of obs = 105
F( 8, 96) = 61.12
Prob > F = 0.0000
R-squared = 0.8705
Root MSE = .1554

| Robust
lnkilled | Coef. Std. Err. t P>|t| [95% Conf. Interval]
lngdppc | -.2491631 .1449582 -1.72 0.089 -.5369029 .0385767
lnpop | .743061 .2831038 2.62 0.010 .1811043 1.305018
war | .0152053 .0384728 0.40 0.694 -.0611626 .0915733
recession | -.0229089 .0628861 -0.36 0.716 -.1477367 .101919
prohibition | .2440428 .0652731 3.74 0.000 .1144769 .3736088
warondrugs | -.0932957 .0545482 -1.71 0.090 -.2015729 .0149816
civilunrest | .0744577 .0436061 1.71 0.091 -.0120997 .161015
lnkilled |
L1 | .6161288 .0773104 7.97 0.000 .4626689 .7695888
_cons | -4.680011 1.918276 -2.44 0.017 -8.487758 -.8722646

The results suggest that there is positive autocorrelation in the data. For every innovation in officers killed in a given year, approximately 62% of that innovation is carried forward into the next year.

The income elasticity is negative, although less than one, and is distinguishable from zero at the 8% significance level. In other words, this relatively simple regression suggests that we have a 92/100 chance that the value is different from zero. If we can agree that this is good enough for econoblogometrics, I would conclude that killing cops (in the United States over the last 100 years) seems to be negatively correlated with income, all else the same.

The population elasticity is positive - as population increases so too does the number of police killed. The elasticity is distinguishable from zero although it is not distinguishable from one.

During war time and recessions there is no greater number of police killed. However, during the years of prohibition there were approximately 24% more police killed per year, compared to non-prohibition years. During the years of civil unrest in the 1960s approximately 7% more police were killed per year. Finally, during the years of the "war on drugs" there have been fewer police killed relative to the earlier years of the twentieth century (I wonder why that is - perhaps drugs in the late 20th century isn't as thugs vs. cops as early 20th century rum running? Perhaps the violence in the drug industry is thugs vs. thugs?).

Next come a few diagnostics:
. durbinh, force

Durbin-Watson h-statistic: -.0700779 t = -.4793672 P-value = .6328

. whitetst

White's general test statistic : 33.32672 Chi-sq(33) P-value = .4514

. ovtest

Ramsey RESET test using powers of the fitted values of lnkilled
Ho: model has no omitted variables
F(3, 93) = 0.56
Prob > F = 0.6415

. predict res, r

. runtest res , thresh(0)
N(res <= 0) = 51
N(res > 0) = 55
obs = 106
N(runs) = 52
z = -.38
Prob>|z| = .71

The Durbin's h-test (used instead of the Durbin-Watston test because of the lagged dependent variable in the model) suggests no correlation (p-value well above 0.10). Likewise, the White test for heteroscedasticity, which is not expected to reveal anything as we are working with time series data, shows no problems with heteroscedasticity. The OVTEST is a version of the Ramsey RESET test for model misspecification/omitted variables. The test fails to reject the null hypothesis of model misspecification (p-value of 0.64). Finally, the Runs Test tests whether the residuals of the regression systematically distributed rather than random. Again, the null hypothesis of randomized errors is not rejected (p-value 0.71).

Is this analysis technically savvy enough to pass muster at a high quality economics journal? Probably not, but that wasn't the intention. From initially seeing the story of 150 police killed in the U.S. in 2006 to my typing this sentence has been approximately 30 minutes. If we believe the accuracy of the parameters we have estimated, then our economic intuition is supported.

I think econoblogometrics (blogging econometrics) should be at the same basic level as undergraduate econometrics. In both cases, interesting insights can be obtained without a lot of fancy theory or super high-powered statistics and empirical techniques that require a Ph.D. to even begin to understand. Moroever, what takes me thirty minutes might well take an undergraduate econometrics students a couple of days, if not weeks.

Econoblogometrics - you saw it here first.

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