# Estimating time invariate variablies in difference-in-differences with fixed effects

I am using the following fixed effects model that has intercept and slope coefficient that vary across individuals.

$$Y_{it}$$ - dependent vector variable for monthly income for individual i at time t.

$$COVID_i$$ - dummy equal 1 if the interview took place on or after the enforcement of the first lockdown policy in the UK and 0 prior to that.

$$Eth_i$$ – categorical variable indicating individuals ethnicity

$$φ_i$$ - individual fixed effects - everything that varies across individual but doesn’t vary across time

$$δ_t$$ – year fixed effects – anything that only varies across time. δ_t and COVIDi are not perfectly corelated because one measures the year effect while the other is down to a day level – COVID is picking the effect over and above the dummy variable for 2020.

$$β_1$$ - estimates the change in income as a response to the pandemic

$$β_2$$ - captures how income changes at time t for individual i when i is of a specific ethnicity. Note that the individual fixed effects, 𝜑i, absorb the ethnic indicator as it doesn’t vary over time

$$β_3$$ - captures the differential post-pandemic effect on Y_it, for individuals from non-white ethnicity for example, relative to those from white ethnicity

($$COVID_t×Eth_i$$) - interaction term between ethnicity and post first COVID-19 lockdown which enables for difference in difference analyses.

$$ε_{it}$$ is the error term which can be both cross sectionally and serially correlated.

My questions is can I estimate the interaction term in fixed effects model even if the independent variable of interest doesn’t vary over time? In other words, I know I can not estimate β_2 but can I still estimate β_3 ?

My questions is, can I estimate the interaction term in a fixed effects model even if the independent variable of interest doesn’t vary over time?

The "variable of interest" should be the interaction of $$Eth_i$$ with $$COVID_t$$. The product term produces a variable that varies both across units and over time, and thus shouldn't be dropped. You can actually safely omit the individual main effects, as doing so doesn't affect your estimate of the treatment effect. But let's unpack that a bit.

In the presence of the fixed effects, the constituent terms cannot be estimated independently. However, it shouldn't impact your estimate of $$\beta_3$$, as the relevant information is already captured by the fixed effects. The "ethnicity" variable is a fixed characteristic of individuals, and is thus collinear with the individual fixed effects. Note how membership to a "treatment group" is time-constant as well; you're either in the exposed group or you're not. Membership to the treatment/control group is just one more fixed attribute. Adjusting for time-constant factors is completely meaningless in the presence of the individual fixed effects. Similarly, the lockdown dummy varies over time but is common across all individuals, and is thus collinear with the year fixed effects.

In short, you can estimate your model as is, just don't expect software to return estimates for $$\beta_1$$ and $$\beta_2$$.

### Concerns

1. $$Eth_i$$ should equal 1 for the treatment group, 0 otherwise. Presently, you state that it's a categorical identifier. I'm not sure that this is what you want. It appears you want to assess some sort of differential response to the pandemic between Whites and non-Whites. If so, then $$Eth_i$$ shouldn't enter the model as separate dummies for all the different ethnicities. It should clearly index the individuals $$i$$ in the "exposed" group. I suspect everyone experienced some sort of economic strain post-lockdown, but maybe one group was more sensitive to the resulting policy. You need to clarify this for the readers of your study.

2. I cannot reliably discern the time frequency of your data. For example, $$Y_{it}$$ is income for individual $$i$$ in month $$t$$; this suggests you have monthly data. Later in your post, you indicate that you have observations at the "day level"; this suggests a daily time frequency. To confuse matters even more, your model includes the parameter $$\delta_t$$, which denotes "year" fixed effects. I can't say with my experience what frequency your working with, but I suspect your outcome varies by month—not year. Assuming you do, in fact, have monthly data, then you should be estimating month fixed effects. This is not something I would overlook, especially if you want to exploit the "timing" of the intervention. I believe the first nationwide "stay-at-home" order was issued in the last week of March in 2020. In this setting, the post-COVID indicator should "turn on" (i.e., switch from 0 to 1) in April, the first full month of lockdown. Note how you couldn't exploit the precise timing of the nationwide policy with yearly observations. The pre-/post-COVID indicator would switch from 0 to 1 in 2020 onward, even though the effects of the policy really started to 'kick in' during the second quarter of 2020.

After perusing the comments, I discovered that you're using the Understanding Society UK Household Survey. Household participants are surveyed once per year, so we can't exploit the pre-/post-pandemic change at a much finer time frequency. It would be desirable if we could acquire responses on a quarterly basis, but that's not as easy as it seems, not to mention quite costly. To be clear, I don't know how you could exploit the timing of the policy by day. Once a survey is completed, a respondent's next interview is usually one year from the date of the previous interview. What you need is a pre-/post-pandemic time indicator for each participant, which should clearly delineate whether a survey wave is before or after the lockdown policy. Software should return four difference-in-differences estimates, where the economic effects of lockdown among the ethnic subgroups are now interpreted as relative to White respondents.

A last word with respect to your interpretation of the coefficient on the time indicator. This is the expected mean change in your outcome among White respondents. However, given that you're working with a balanced panel and a small number of survey waves, I would consider reporting the coefficients on the individual year dummies. Those effects represent the expected mean difference in your outcome in some year $$t$$ relative to a particular reference year, which is likely going to be the first survey year.

• Dear Thomas Bilach, thank you very much for your timely response it proved invaluable for my research. I am using the Understanding Society Survey, which has a few ethnicities listed; thus in Stata, I use 'i.eth' in my regression. Is this sufficient or should I create a dummy for each ethnicity? Apr 22, 2022 at 11:52
• Individual interviews are recorded down to the exact day. However, the frequency of interviews is roughly every year. In my current model, the dummy for COVID becomes 1 at exactly 23rd of March 2020 while the time fixed effect was yearly – this is why Stata could estimate it. However, as you noted, the effect of nationwide lockdowns should really kick in in the second quarter of 2020. I could simplify the data down to quarter level and have δt to be quarter fixed effects, but I'm unsure how to go about this when my observation frequency is every year. What would your recommendation be? Apr 22, 2022 at 11:52
• Please see stats.meta.stackexchange.com/questions/6304/my-upvoting-policy, when you find a question sufficiently clear to write an answer, consider to upvote the question! Apr 22, 2022 at 16:36
• @MilkoHristov Yes, Stata will ‘dummy out’ the individual categories, but that is not what you want. Which ethnicity are you interested in? Apr 24, 2022 at 15:34
• I am interested in White, Black, Arab, Mixed and Asian. Currently I have White=0, Black=1, Asian=2, Mixed= 3 and Arab=4. Why would that approach not be appropriate? Apr 24, 2022 at 20:03