Difference-in-difference regression with pooled cross sectional data - fixed effects

Question

I'm trying to estimate a difference-in-difference model with pooled cross-sectional data. The dataset consists of rental prices in one city for the years of 2011-2021 and includes a wide range of other variables (e.g. living size, lift, heating etc.).

I added a dummy variable "braked": 0 if the dwelling is in the control group and 1: if the dwelling is in the treatment group plus a time dummy variable "rba": 0 if the policy (rental brake in Germany) is not active and 1: if the policy is active at the time of observation (all units are treated/not treated at the same single point of time)

The model is specified as:

model_advanced <- lm(rent_log ~ construction_year + living_space_sqm
                                      + floor_of_object + number_of_floors 
                                      + elevator + balcony + built_in_kitchen + garden + cellar
                                      + Sophisticated + Normal + Deluxe + Simple
                                      + floor_heating + selfcontained_central_heating
                                      + district_heating + gas_heating + oil_heating
                                      + night_storage_heaters + electric_heating
                                      + well_kempt + like_new + completely_renovated
                                      + modernised + reconstructed
                                      + needs_renovation + dilapidated
                                      + rba + braked + braked*rba,
                                      data = data_rbind1)

My questions are the following:

1. I have to include the location of the dwellings in some way, because the rental prices are of course higher in the center of the city than outside of the city. Could I just add a municipality fixed effect (based on the zip code)?
2. Should I create a time fixed effect for the years 2011-2021 to account for changes within time? (see in the following code: variable "ajahr" is the date of offer of the rental unit)
3. Could the results be biased if the number of observations in the treatment and control group differ? I have

My R Code looks like the following with fixed effects:

model_advanced_FE <- feols(rent_log ~ construction_year + living_space_sqm
                                      + floor_of_object + number_of_floors
                                      + elevator + balcony + built_in_kitchen + garden + cellar
                                      + Sophisticated + Normal + Deluxe + Simple
                                      + floor_heating + selfcontained_central_heating
                                      + district_heating + gas_heating + oil_heating
                                      + night_storage_heaters + electric_heating
                                      + well_kempt + like_new + completely_renovated
                                      + modernised + reconstructed
                                      + needs_renovation + dilapidated
                                      + rba + braked + braked*rba| zipcode + ajahr,
                                      data = data_rbind1,
                                      demeaned = TRUE
                                      )

Would this be the right approach?

Answer 1

In order for this to work, a sampled rental/dwelling unit $i$ must belong to a particular group, say group $T_{g} \in {0,1}$ (1 = braked), and the two groups should be observed before and after the reform. If my understanding is correct, newly built or modernized/renovated apartments serve as part of the unbraked group; they are not affected by the rent brake (i.e., treatment). The other newly advertised rentals/dwellings serve as part of the braked segment; they're the units that are actually affected by the reform. Your goal is to follow these two groups pre-/post-reform, in keeping with a traditional difference-in-differences framework. The model would look something like the following:

$$ \log(R_{igt}) = \gamma T_{g} + \lambda P_{t} + \delta (T_{g} \times P_{t}) + X_{igt}'\theta + \epsilon_{igt} $$

where $R$ is the rental price (logged) of interest. As indicated before, $T_{g}$ a treatment dummy, grouping units into either the braked/unbraked group. $P_{t}$ is a time indicator, where a value of 1 means we are in the reform epoch, 0 otherwise. The product of these two terms returns the difference-in-differences estimate. And lastly, the covariate vector $X$ is whatever housing characteristics you want to include in your model; it seems like you have a lot of them.

I have to include the location of the dwellings in some way, because the rental prices are of course higher in the center of the city than outside of the city. Could I just add a municipality fixed effect (based on the zip code)?

The names of the different areal units (i.e., territorial divisions) may vary, but yes including a set of district dummies is one way to account for location. You could easily estimate them in a linear model. I recommend excluding the coefficients from your tabular results as your model already includes a large number of covariates. Try estimating the above model with, and without, the district dummies and assess how they affect your point estimates.

Also, if you're concerned about where a unit is relative to the center of a jurisdiction, then consider including a covariate which represents the distance to the city center, which you should consider including logarithmically.

Should I create a time fixed effect for the years 2011-2021 to account for changes within time?

Yes.

But note that the pre-/post-reform indicator $P_{t}$ is enough to capture the time effects. It's just another way to capture those common time shocks affecting both groups. In a setting where the reform (i.e., treatment) affects the "braked group" at a precise, well-defined point in time, then you'll find the point estimates shouldn't change much whether you add a pre-/post-reform indicator or include the individual year dummies.

If you do decide to estimate a series of year effects, then you'll discover that software will invariably drop the pre-/post-reform indicator. In the research you cited, tabular results don't include a main effect for "post" (see, e.g., Table 2). The year dummies completely absorb the pre-/post-reform indicator. Again, a pre-/post-reform indicator is just another way of modeling the time shocks affecting both groups.

Could the results be biased if the number of observations in the treatment and control group differ?

It could, but I assume your sample size if sufficient. In previous evaluations of the rent brake, the number of sampled $i$ was usually around 10,000 units. In some city jurisdictions (e.g., Berlin), the number of rentals was nearly seven times that amount.

I included some R code below, which explicitly introduces greater model complexity as we include more and more adjustments.

## Unadjusted difference-in-differences

model_1 <- lm(rent_log ~ braked*rba, data = ...)

## Adjusted difference-in-differences
 # covariates = housing characteristics (e.g., year of construction, number of floors, etc.)

model_2 <- lm(rent_log ~ braked*rba + covariates, data = ...)

## Adjusted difference-in-differences
 # as.factor(zipcode) = district (location) fixed effects

model_3 <- lm(rent_log ~ braked*rba + covariates + as.factor(zipcode), data = ...)

## Adjusted difference-in-differences
 # as.factor(year) = year (time) fixed effects

model_4 <- lm(rent_log ~ braked*rba + covariates + as.factor(zipcode) + as.factor(year), data = ...)

To be honest, the feols() function from the fixest package offers some great functionality. The fixed effects listed on the right-hand side of the | shouldn't appear in your output, which is nice. Just note that rba will be dropped in the presence of the year fixed effects. I am somewhat partial to the latter approach, as it will result in somewhat cleaner output.