Interpreting a log-level model: Dummy is switching 'on' and 'off' over time How does the interpretation of the coefficient on a dummy variable change in settings where the indicator is switching 'on' and 'off' repeatedly over time? Suppose I have complaint data on hotels $i$ over days $t$. Some of the hotels employ guards, while others do not. Interestingly, security is only present in and around the treated facilities two days per week. The other hotels have no coverage at all.
The model is taking the following form:
$$
\text{log}(y_{it}) = \beta d_{it} + \gamma_i + \lambda_t + u_{it},
$$
where $d_{it}$ is equal to 1 if a hotel is assigned guards and only during the days when they are actually surveilling the location. The parameters $\gamma_i$ and $\lambda_t$ represent hotel and day fixed effects, respectively. This framework mirrors the more general difference-in-differences estimator.
Here is what I know about the interpretation of a dummy variable when the outcome is log-transformed:

*

*If $d_{it}$ switches from 0 to 1, the % impact of $d_{it}$ on the outcome is: $100 \times (\exp(\hat{\beta})-1)$

*If $d_{it}$ switches from 1 to 0, the % impact of $d_{it}$ on the outcome is: $100 \times (\exp(-\hat{\beta})-1)$
But the intervention dummy is switching back and forth somewhat arbitrarily. Hotels receive this 'on' and 'off' coverage over the entire year. If this helps, see the abridged data frame below (only hotel 2 receives intermittent coverage):
$$
\begin{array}{ccc}
hotel & day & d_{it} \\
\hline
 1 & 1 & 0  \\
 1 & 2 & 0  \\
 1 & 3 & 0  \\
 1 & 4 & 0  \\
 1 & 5 & 0  \\
 1 & 6 & 0  \\
 1 & 7 & 0  \\
\hline
 2 & 1 & 0  \\
 2 & 2 & 0  \\
 2 & 3 & 0  \\
 2 & 4 & 1  \\
 2 & 5 & 0  \\
 2 & 6 & 0  \\
 2 & 7 & 1  \\
\end{array}
$$
This estimator is often described as averaging all the two-by-two difference-in-differences estimators. It is my understanding that the estimate of $\beta$ is the average of all the sub-estimators when there is a change from 0 to 1.
Question: Does the "percentage" interpretation need to account for hotels moving in and out of the treated condition repeatedly? As shown, hotels 'switch out' of the treated condition multiple times.
 A: This is a complicated question. I will side-step how well DID works in a setting with both staggered/differential timing and transient/spikey/pulsing/non-absorbing/reversible treatments. There is a nice summary of this growing literature here. The short summary is that the weights that you mention can get really weird and even turn negative.
Another relevant paper is Imai, Kim, and Wang (2018), where democracy and war are the treatments.
Now for your main question. I think the reason you have asymmetry here is that
percent changes are not symmetric. Say $\hat \beta_D = 0.6433$. That means when D goes from 0 to 1 you would expect $$\exp(0.6443)-1= .9047,$$ which is $+90\%$. So if the baseline was 50 incidents per month, you would now expect 95. Now suppose D goes from 1 to 0. Then $$95 \cdot (\exp(-0.6443)-1)=95 \cdot (-.47497) = -45,$$ so that gets us back down to 50 from 95. The percent change factor is different in magnitude from before since the baseline is now higher.
You can also reduce small sample bias by calculating
$$\exp (\beta_D - \frac{1}{2}\hat \sigma^2_{\hat \beta_D})-1.$$
