# What is the best way to model when the treatment is intermittent across years?

Let's say the goal is to gauge the effect of deploying certain clean technology (dummy variable: 1 for the year the technology is deployed, 0 otherwise) on a firm's pollution level (continuous variable, yearly average). The dataset is an unbalanced panel with roughly 100 different firms in more than 30 cities across 10 years. Not all firms will deploy the technology (treated) during the time period, and probably the most complex thing in the dataset is that the technology deployment (treatment effect) is intermittent in various ways for different firms.

For example, Firm A may introduce the technology in Year 3 through Year 5, and then, for some reasons, it did not use the technology for Year 6 and Year 7, but the technology is re-deployed again from Year 8 and the rest of years. Firm B could have a totally different pattern in terms of the timing of deploying the technology, etc.

To achieve the goal, I think a straightforward approach would be to model the deployment effect directly using a fixed-effect model with controls for both Firm and Year:

$$Pollution = \alpha + \beta_1 Tech\ + \omega_i Controls + \gamma_i Firm \ + \theta Year + \epsilon$$ The controls in the above equation refer to all possible firm-level controls varying across years(e.g., revenue, debt, number of employees, etc..)

My first question is: In this case, is it legit to claim that the estimates on $$\beta$$ are the average treatment effect (ATE) of deploying the technology on a firms' pollution level?

Then, I saw some people use the so-called pooled-OLS to model similar datasets, but using controls at a larger scale (i.e., city-level rather than the firm-level):

$$Pollution = \alpha + \beta_1 Tech\ + \omega_i Controls + \gamma_i City \ + \theta Year + \epsilon$$

My second question is: is it legit to use the pooled-OLS model with a broader level of controls for this case, especially considering the treatment effect is intermittent? How to interpret the estimate on $$\beta$$ then?

I have never used the pooled-OLS before and I do not fully understand why it would work and the interpretation of the estimates.

• There are a few ways to go about this. But first, what do you mean when you say broader level of controls? Do you mean controls at the city or regional level? Does it matter what city the firm is in with respect to treatment assignment? Jul 17 '20 at 1:03
• Hi @ThomasBilach , yes, with the broader level, I mean the city/regional level fixed effects. Anything that larger than the firm level fixed effect. We can consider the technology deployment is voluntarily and we can, for now, ignore the correlation with the city where a firm locate. But, since the pollution level may also change due to other reasons within that city (say an environmental policy changes), we may want to think about the effect from city level in the model. Jul 17 '20 at 1:10
• Are the firms of comparable size, pollution levels comparable? Otherwise, maybe some multiplicative model with an offset for size? Jul 18 '20 at 18:42
• @kjetilbhalvorsen yes! We can assume the firms are similar and same type of polluting firms for now. Actually, even if there are some differences in the firm, I can control them in the model, such as adding the company size, industry, etc. the pollution level may vary due to firm size; but we can standardized the pollution level by the firm’s size (divided pollution level its annual revenue for example). Those issues are relatively easy to handle, I think. Jul 18 '20 at 18:47

You can go about this in a couple of ways. It appears you want to estimate the following equation:

$$\text{Pollution}_{ict} = \gamma_{c} + \lambda_{t} + \delta \text{Tech}_{ct} + X_{ict}’ \eta + \epsilon_{ict},$$

where you observe $$i$$ firms within $$c$$ cities across $$t$$ years. $$\gamma_{c}$$ and $$\lambda_{t}$$ are fixed effects for cities and years, respectively. Your treatment dummy $$\text{Tech}_{it}$$ is at the city-year level. But, it appears the treatment is not well-defined at this higher level of aggregation, so we can proceed with the following:

$$\text{Pollution}_{it} = \alpha_{i} + \lambda_{t} + \delta \text{Tech}_{it} + X_{it}’ \eta + \epsilon_{it},$$

where $$\alpha_{i}$$ represents firm fixed effects. Don't worry about the city effects from earlier; the firm effect will absorb the city effect. $$\text{Tech}_{it}$$ is your policy (treatment) dummy. This is a two-way fixed effects estimator. It is often referred to as a 'generalized' difference-in-differences (DD) equation. See this post for more information.

In your setting, some firms receive a treatment and others do not. Some firms even move into and out of a treated condition multiple times. Thus, while some of the treated firms have one exposure, others may have more than one, and not all firms exhibit the same treatment history. Your treatment dummy in this case should reflect reality as close as possible. In other words, $$\text{Tech}_{it}$$ should equal 1 for treated firms and only during years when the technology is implemented, 0 otherwise. Note, the pattern of the treatment dummy should reflect all 'on' and 'off' periods of treatment. As for those firms never espousing the new technology, they should be coded 0 for the entire observation period. Again, $$\text{Tech}_{it}$$ is not indexing a treatment group; rather, it 'turns on' only in the 'firm-year' technology adoption periods, 0 otherwise! As a simple binary indicator, $$\text{Tech}_{it}$$ makes no distinction between groups of firms treated once or those treated more than once. You're either treated at time $$t$$ or you're not. Imagine a column of 0's for all firms and years. Set any firm-year cell equal to 1 if the technology is being used in that firm-year, 0 in all other cells. Your estimate of $$\delta$$ is your treatment effect.

This equation can be estimated via ordinary least squares. Simply regress $$\text{Pollution}_{it}$$ on $$\text{Tech}_{it}$$ and a series of dummies for all firms and all years. I addressed a similar post recently regarding the coding of the treatment dummy in these settings.

• Hi @ThomasBilach, THANK YOU for your input. I like the way of using "turning on/off" rather than just a treatment effect to interpret the technology variable. It makes sense to me. In my case, there is no city-wise policy for technology adoption, all are voluntary decisions by individual firms. I fully agree with your 2nd equation. My only confusion now is that is it possible (still correct?) to interpret the result if I stick to using a 'city' level fixed effect (your 1st equation) for my case when the city-wise treatment is not clearly defined? Jul 20 '20 at 0:09
• Just so my answer is clear, what do you mean by “turning on/off” as my interpretation of the technology variable? You can code the treatment dummy flexibly in this setting. Firms move into and out of the treatment condition, and your policy indicator should match on/off periods of treatment. This approach can handle intermittent treatment exposure. The coefficient on $\delta$ is your treatment effect. I should ask: is treatment at the firm level or city level? If treatment is at the city level, then all firms citywide would, or should, be treated. Is this the case? Jul 20 '20 at 17:08
• Hi @ThomasBilach. I was confused if coding as 0/1 (turning on and off) intermittently is legit since a normal DD design would have a continuous status once treated. But your other post about a similar issue is helpful. As to my case, the treatment is at the firm-level and there is NO city-level treatment. So you can see some firms are treated while others are not within a city. In this case, is using the city-level fixed effect still correct? (my guess is not since I don't know how to interpret the result then, but I have seen people doing it, that's my concern). Jul 20 '20 at 17:18
• I just want to see if other folks will have some other thoughts about my questions. But I will mark your answer as the correct one if no others show up later. THANK YOU VERY MUCH! Jul 20 '20 at 17:20
• The policy dummy can accommodate irregular patterns. In the other post, the OP quotes from Jeffrey Wooldridge's text, which indicates that this dummy can take on any pattern. In your case, treatment is at the firm-level, so I recommend using the latter equation. If you opt to go with a treatment dummy at the firm level and use city and/or year fixed effects, it will return a different estimate of the treatment effect. I hope this helps. Jul 20 '20 at 18:21