# Interpreting coefficients with Delta ln as independant and dependent variables

Imagine we have a DGP such that

$$\Delta \ln y_{it}=\beta\times\Delta\ln x_{it}$$

How do you interpret the $$\beta$$ coefficient, since it is expressed in an approximation of a growth rate?

$$\ln(y_{it} / y_{it-1}) = \beta\ln(x_{it} / x_{it-1})$$
So it becomes a log-log regression equation. In this form, it means that if we change the ratio of $$x_{it}$$ to $$x_{it-1}$$ by one percent, the ratio of $$y_{it}$$ to $$y_{it-1}$$ will change by $$\beta$$ percent, on average. In other words, if $$x_{i}$$ grows by one percent between two periods, $$y_{i}$$ is expected to grow by $$\beta$$ percent in that same time span.