Why do the coefficients of cross-sectional fixed effects and time fixed effects become zero?

I currently have a panel data set that contains the quarterly increment of loans initiated from more than 300 cities in China over the period from 2011Q1 to 2020Q2. I want to examine the impact of COVID-19 on lending activities. I believe that both time fixed effects and city fixed effects exist—both the F test for poolability and Hausman Test support a fixed-effect model.

I use the panel procedure of SAS to analyze it. I also set a dummy variable, "event", to indicate whether the pandemic has occurred. The dummy variable "event" equals to 1 if the period is 2020Q1 or 2020Q2. Currently, I have not added any interactive terms. If I specify a one-way model using fixone option that only allows cross-section fixed effects, I found that some coefficients of these fixed effects are significant while some are not. The coefficients of the dummy variable "event" and other control variables are also significant. By now, it is acceptable.

However, if I specify a one-way fixed effects model using fixonetime option that only allows time fixed effects, it will be a mess. Although the coefficients of all time fixed effects are significant, the coefficient of the dummy variable "event" becomes zero. I am wondering whether it is caused by perfect multicollinearity. After all, in my setting, event = 1 means the same thing as the fixed effects of 2020Q1 and 2020Q2. My guess is that the impact of the dummy variable "event" has already been absorbed by time fixed effects. Is that right? If it is correct, does it mean that you can never use the time fixed effects and some dummy variables indicating certain event happens at the same time? However, several papers are studying the impact of COVID-19 using them simultaneously in their model. They did not mention any problems of such a setting. So I feel confused.

In a branched setting, instead of using the dummy variable "event", I use two dummy variables, "event_level1" and "event_level2", which equals to 1 if the period is 2020Q1 and 2020Q2 respectively, to visualize the dynamics of COVID-19 impact. The result is getting worse because the coefficients of all fixed effects and interested dummy variables "event_level1" and "event_level2" become zero this time. It looks caused by other reasons instead of multicollinearity this time. Are there any generous and smart friends who know the reason? Thank you so much!

• Welcome. Is this event dummy standardized across all cities? I assume so, but it is not clear. In other words, does it ‘turn on’ in Q1 and Q2 in all 300+ cities? – Thomas Bilach Sep 16 '20 at 19:11
• Since COVID-19 is a major and comprehensive shock, I think it's reasonable to assume that this event dummy works across all cities. @Thomas Bilach – StudentM Sep 17 '20 at 0:47

Collinearity appears to be the problem.

I currently have a panel data set that contains the quarterly increment of loans initiated from more than 300 cities in China over the period from 2011Q1 to 2020Q2. I want to examine the impact of COVID-19 on lending activities.

SARS-Cov-2 is assumed to affect the lending practices of all cities within China at precisely the same time. Your event dummy 'turns on' in all cities in the first and second quarters of 2020. Note, your event dummy has the same 'within-city' pattern. Your first model should look something like this:

$$\text{Lending}_{cq} = \beta_{0} + \beta_{1}\text{Event}_{q} + \alpha_{c} + \epsilon_{cq}$$

where you observe the lending activities in city $$c$$ across quarters $$q$$. $$\text{Event}_{q}$$ is $$q$$-subscripted. The dummy exhibits no variation across cities. Demeaning your equation (or estimating dummies for all cities) 'sweeps out' $$a_c$$ (i.e., the 'city' effect). This model is estimable.

However, adding time (quarter) fixed effects introduces collinearity concerns. Suppose you have quarterly data on 300 cities from 2011 onward; I assume your panel ends in the second quarter of 2020. Now say you wish to estimate a two-way fixed effects estimator using the least-squares dummy variables (LSDV) approach; this is algebraically equivalent to estimation in deviations from means. This should result in 300 separate city effects and 42 separate quarter effects. The quarter fixed effects represent the common shocks, absent the pandemic, affecting all cities. By construction, your event dummy is collinear with your last two quarter effects.

My guess is that the impact of the dummy variable "event" has already been absorbed by time fixed effects. Is that right?

Precisely. Again, $$\text{Event}_{q}$$ is equal to 1 in all cities in the first and second quarter of 2020. If you include a model with quarter fixed effects, then you have a unique 2020-Q1 dummy and a unique 2020-Q2 dummy specific to all cities. In other words, $$\text{Event}_{q}$$ is the sum of the last two quarter effects. To break the collinearity, SAS dropped your event dummy. In most software packages, the order of your variables matters inside of a regression function call. If you want an estimate for your event dummy, then I recommend dropping the last quarter. In R/Stata, for example, if you specified your event indicator before the city and quarter effects, it will default to removing your last quarter dummy (i.e., 2020-Q2). You could do this manually in SAS to obtain an estimate for your event (i.e., exposure/intervention) dummy.

In a branched setting, instead of using the dummy variable "event", I use two dummy variable, "event_level1" and "event_level2", which equals to 1 if the period is 2020Q1 and 2020Q2 respectively, to visualize the dynamic of COVID-19 impact.

As already shown, you cannot estimate the dynamics of exposure with city and quarter fixed effects. Such a model is inestimable. Your event dummies are perfectly collinear with your last two quarter effects. Thus, a two-way fixed effects estimator will fail in this setting. Your event indicators must vary over time and across units to use this estimator.

In order to estimate city and quarter fixed effects, you require either a ‘staggered’ exposure (i.e., different cities become exposed at different times) or a setting where a subset of cities were never treated. Because SARS-Cov-2 is a population-level exposure affecting all cities/provinces in precisely the same time periods, then quarter fixed effects will completely absorb your event dummy.

You can safely estimate your model using one-way city-level fixed effects or pooled ordinary least squares.

• Although I said before that the COVID-19 is a comprehensive shock, it is still possible to find such untreated cities if I use the standard "whether the city is locked down" because the lockdown periods of different cities are different. For example, when most China cities reopen, cities in Hubei province, especially Wuhan, are still in lockdown status. Then a typical difference-in-difference analysis with Post * Treated term can be conducted. – StudentM Sep 17 '20 at 1:09
• You can proceed that way. However, if the lockdown periods vary across cities then you should proceed with the “generalized” difference-in-differences equation. I recommend this because your $Post_t$ period wouldn’t be standardized across cities/provinces. – Thomas Bilach Sep 17 '20 at 1:13
• You give me some hints that I can work around multicollinearity by making the event dummy interacted with other variables, like the treated dummy or control variables, instead of including a pure event dummy. So by now, I think out two possible strategies. – StudentM Sep 17 '20 at 1:20
• First, I can use a one-way fixed model that only integrates city fixed effects. In such a model, I can use the event dummies and interact them with other variables freely. I may further include some time-point dummies to hold structural breaks in my panel dataset as long as there is no multicollinearity among these dummies. Second, I can adopt a two-way fixed effects model and only include interaction terms of the event dummy with other variables. How do you think about these two strategies? – StudentM Sep 17 '20 at 1:23
• My answer does not suggest interacting your event dummy with time-varying covariates to break the collinearity. You either have to drop the quarter effects entirely or model the passage of time in a different way. Interacting your event dummy with covariates still results in your main policy dummy (i.e., $\text{Event}_{q}$) being dropped from the model. Have you considered collecting more data from different countries in other regions? Maybe you could exploit variation in treatment timing. Also, maybe there is anticipatory behavior to model. Do lending practices change in late 2019 at all? – Thomas Bilach Sep 17 '20 at 17:28