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?

Thanks for your help


Your equation can be re-written (through the properties of logs) as:

$$ \ln(y_{it} / y_{it-1}) = \beta\ln(x_{it} / x_{it-1}) $$

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.


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