Interpretation of log-level difference-in-differences specification When I run a standard difference in differences specification with a log-transformed dependent variable like:
$$\log(Outcome_{it}) = \beta_1 + \beta_2Treat_i +\beta_3Post_t +\beta_4(Treat\times Post)_{it} +\varepsilon_{it}$$
How do I interpret the coefficient $\beta4$?
Normally in log-level models, I would use the approximation $\%\Delta y= 100\beta_4\Delta x$, but this approximation is only valid for small changes in $x$ (and small $\beta$). In my case $x$ is a dummy variable and as such either $0$ or $1$ ($Treat\times Post$). Is this change considered to be 'small' or do I have too use  $\%\Delta y=e^{\beta-1}$
for interpreting the coefficient?
 A: You can should treat the interaction variable as a dummy and follow this advice from David Giles:
If $Treat\cdot Post$ switches from 0 to 1, the % impact on $Y$ is $100 \cdot (\exp(\beta_4 - \frac{1}{2} \hat \sigma_{\beta_4}^2)-1).$
A: What's required is that $\beta \cdot \Delta x$ be small.  If you know that $\Delta x$ is 1, then that means that $\beta$ has to be small.  How small?  The true proportionate change in $Outcome$ when the dummy rises by $1$ is $\exp(\beta)-1$.  The approximate change is $\beta$. The error from the approximation is:
\begin{equation}
\textrm{Error} = \exp(\beta)-1-\beta
\end{equation}
For small $|\beta|$, this is pretty small.  For example, for $\beta=0.1$ (approximate 10% change), the true percent change in $Outcome$ when the dummy turns on is 10.5%.  Given the usual standard errors in empirical work, I'm happy to ignore this.  By the time you get to a $\beta$ of 0.2 (approximate 20%), the true percent change is 22%.  Willing to ignore this much approximation error?  Again, I am, but you may not be.  This is now a 10% approximation error.  By the time you get to $\beta=0.3$, the true percent change in outcome is 35% rather than 30%, and I am not happy to ignore this any more.
So, my rule of thumb is to ignore this approximation error for $|\beta|<0.2$ and worry about it for $\beta$ bigger than that.
