# How to interpret this regression coefficient?

I am performing a simple single variate regression on the variables crime rate (denoted by crrate) and the probability of getting arrested (denoted by prarrest). To be precise, the variables are defined as:

1. crrate: ratio of the number of crimes committed to county population
2. prarrest: 100* the ratio of arrests to offenses.

The slope coefficient value I got when I regressed crrate on prarrest is -0.067. How can this coefficient be interpreted?

My attempt:

Ceteris paribus, one percentage point increase in prarrest causes the number of crimes committed per 1000 county residents to go down by 67.

Is this interpretation correct? If not, how can it be improved?

• The slope is 0.067 or -0.067? Commented Jul 15, 2020 at 16:08
• @nrivera Sorry, it is -0.067 Commented Jul 15, 2020 at 16:23
• It's not clear to me that interpreting this regression makes any sense. You are including crimes on both sides of the equation (one side called offenses in the denominator and the other side called crimes in the numerator) so that all else equal a change in crime would imply a negative relationship which is scaled by the other variables. So I am not sure it makes much sense to interpret the coefficient until you address the endogeneity issue. Commented Jul 16, 2020 at 2:39
• @AndrewM Wow, thanks for pointing that out. It seems like I need to work on my model a bit more. You have an excellent point. Commented Jul 16, 2020 at 21:57

Assuming that from the beginning you made the regression with prarrest with transformation *100 (say you have for example [0.012,0.093] and you transform it into -> [1.2,9.3], I remark this cause its critical to the interpretation).

Ceteris paribus, one percentage point increase in prarrest causes the number of crimes committed per 1000 county residents to go down by 67.

It is correct, you could se this with this approach (assuming its linear regression), taking into account the your hypothesis must have this from:

$$crrate=\hat{\beta_0}+\hat{\beta_1}(prarrest)\;\;\;\;\;$$where $$\hat{\beta_1}=-0.067$$

$$\frac{\Delta(crrate)}{\Delta(prarrest)}=-0.067 \implies \Delta(crrate)=-0.067\Delta(prarrerst)$$

Now, if $$\Delta(prarrest)=1$$ then $$\Delta(crrate)=-0.067$$.

Since prarrest is in the terms clarified from the beginning, this $$\Delta(prarrest)=1$$ means a whole percentage point increase (e.g. from 34(%) to 35(%)), then looking at the terms or the $$crrate$$ variable (#crimes/population, so for one individual of the population how many crimes are committed), a -0.067 change means a decrease in the order of 0.067 per each person, so multiplying this by 1000, would give us decrease of 67 per 1000 people. Yes, your try is correct.