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I have the following model: $$ Gini_{it} = \alpha_i + \beta_1\ln(BNP_{it}) + \beta_2trade_{it} + \epsilon_{it}, $$ where $Gini_{it}$ is the Gini-index from 0 to 100, $\ln(BNP_{it})$ is $\ln$ of the GDP per capita, and $trade_{it}$ is the sum of import and export divided by BNP. I also have some other control variable, but I am interested in these two.

How do I interpret the estimates of $\beta_1$ and $\beta_2$. Is it correct to say if we increase BNP per capita by one percent, we expect the Gini-index to increase by ($\beta_1$/100) units? And if we increase trade by 1 percent, we expect the Gini-index by $\beta_2$ percent?

Hope someone can help.

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1% increase in BNP:

$\text{Gini} \sim \beta_1 \,ln(\text{BNP})$

If we go from $\text{BNP}_1$ to $\text{BNP}_2$, where $\text{BNP}_2=1.01\,\text{BNP}_1$, $\text{Gini}_2\sim\beta_1\,ln(1.01\,\text{BNP}_1)$, and the difference in $\text{Gini}$ keeping every other factor constant will be:

$\Delta\text{Gini}=\beta_1\,\big(ln(1.01\,\text{BNP}_1)\,-\,ln(\text{BNP}_1)\big) = \beta_1\,ln\Big(\frac{1.01\,\text{BNP}_1}{\text{BNP}_1}\Big)= ln(1.01)\,\beta_1 \sim\frac{\beta_1}{100}$.

So an increase of $1\%$ in BNP, will increase the GINI $\sim 1\%$ of $\beta_1$. As discussed previously, this won't be a linear growth over BNP (actual growth in red):

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1% increase in trade:

$\Delta\text{Gini}=\beta_2\,\big(1.01\,\text{trade}_1\,-\,\text{trade}_1\big) =0.01\,\text{trade}_1\,\beta_2$ or an increase of $1\%$ over the prior contribution of trade: $\beta_2\,\text{trade}_1$ (linear contribution).

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If you take the partial derivatives of $Gini_{it}$ with respect to $BNP_{it}$ and $trade_{it}$ you will get the rates in which $Gini_{it}$ grows if you maintain everything else is constant. This is a good approach to identify what are the roles of your parameters.

$R_{BNP} = \frac{\partial Gini}{\partial BNP}$

$R_{BNP} = \frac{\beta_1}{BNP} $

and

$R_{trade} = \frac{\partial Gini}{\partial trade} $

$R_{trade} = \beta_2$

So $G_{ini}$ is growing faster with higher values of $\beta_1$ and $\beta_2$.

The first equation shows that the rate which Gini-index grows is inversely proportional to the value of $BNP$. This means that the growth rate will change every time $BNP$ changes, and so Gini-index will never grow constantly as you've expected.

The second equations shows us that the rate of growth with respect to $trade$ will always be a constant. But even though Gini-index is linear with respect to $trade$ you cannot assume they will increase by the same amount. You can prove this by making everything constant but your variable to analyse, and giving a large value to the linear coefficient:

$(1.1\times Gini) \neq (1.1\times trade) + 1000$

$(1.1\times Gini) = (1.1\times trade) + C$ is true only if $C = 0$

The same logic can be applied to the first equation. Even if $BNP = 100$ you won't get a ratio of $1:\frac{\beta_1}{100}$.

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You are on the right lines but it is $\ln(BNP)$ which increases by a unit to effect a change of $\beta_1$ in Gini. So that corresponds to multiplying BNP by $e$ (=2.71828).

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