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This might be a basic question. The article given below checks the relationship between crime and income inequality. https://www.sciencedirect.com/science/article/pii/S0165176508001110. Both crime and gini is log transformed.

Table 1 shows -0.6942 as the coefficient for regressing crime on income inequality. There are other values for different types of crime but let's focus on violent crime in Table 1. The interpretation of the results is states as follows "So when Gini ratio increases by 1%, violent crime increase by 0.6942%"! I am not able to understand the interpretation since the coefficient is negative.

My thoughts:

Assume $$y=f(x), \quad 0<x<1 \text{ and } y \text{ is a non-negative real number}$$

if $cov(x,y)=+ve$, then

$$cov(ln(x),y)=E(ln(x)y)-E(ln(x))*E(y)$$

since $ln(x)$ is negative first term is negative. The second term is also negative as $E(ln(x))$ is negative. However, I am not sure whether the outcome is $+ve$ or $-ve$? Is it inconclusive?

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The only thing I can think is that they "backwards-coded" Gini coefficient when running the regression.

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    $\begingroup$ As in, 1 would imply perfect equality, and 0 would imply maximal inequality? $\endgroup$
    – chris jude
    Jan 10, 2023 at 11:49
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    $\begingroup$ That's what it looks like to me. $\endgroup$ Jan 10, 2023 at 13:05
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    $\begingroup$ Thanks for the reply. The lack of information in the article prevents me from verifying. Just to reconfirm: The author mentions in footnote 3, "Due to the log transformation, coefficient of Gini ratio, which is between 0 and 1, has a negative value". Isn't it a conceptually wrong statement? $\endgroup$
    – chris jude
    Jan 10, 2023 at 13:49
  • $\begingroup$ From what I understood, there is a possibility for cov(log(x),log(y)) to be -ve even when Cov (x,y) is +ve; see comments in this query stats.stackexchange.com/questions/601063/… but that in itself is not sufficient or necessary to claim Cov (x,y) is +ve. $\endgroup$
    – chris jude
    Jan 10, 2023 at 13:50
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    $\begingroup$ It is true that $ln(Gini)<0$ because $Gini<1$. But some Stata simulations don't convincee me that this necessarily explains the result. clear all set obs 100 gen x = runiform() gen y = 2*x + runiform() gen lnx = ln(x) gen lny = ln(y) reg lny lnx $\endgroup$ Jan 10, 2023 at 14:31

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