Quarterly regression, year-fixed effects

I am trying to inspect the heterogeneous effects of a USA-based shock on a panel of countries using quarterly panel data. My variables of interest are the USA-based shock (which has only temporal variation and is mildly serially correlated) and the interaction of this variable with another variable that has cross-sectional and temporal variation too. My question would be whether it makes sense to include year-fixed effects in this regression?
I mainly worry about something like controlling for a downstream variable (subtracting part of the effect of the US shock with the year-fixed effects).

I guess there's no easy answer. It depends on what you think the data generating process is.

Let $$y$$ be year, $$q$$ be the quarter and $$c$$ be the country.

Then $$S_{y,q}$$ is the USA shock at year $$y$$ and quarter $$q$$, $$X_{c,y,q}$$ is the observable covariates of country $$c$$, at year $$y$$ and quarter $$q$$ and let $$Y_{c,y,q}$$ be the variable of interest.

From reading your question, your basic regression looks like this: $$Y_{c,y,q} = \alpha_0 + \alpha_1 S_{y,q} + \alpha_2 X_{c,y,q} + \alpha_3 (X_{c,y,q} \times S_{y,q}) + \varepsilon_{c,y,q}.$$ The underlying assumption is here that: $$\mathbb{E}(\varepsilon_{c,y,q}|S_{y,q}) = \mathbb{E}(\varepsilon_{c,y,q}|X_{c,y,q}) = 0$$

• A first thing you might be worried about is that $$Y_{c,y,q}$$ might be (systematically) different depending on the country in addition to what you observe in $$X_{c,y,q}$$. In this case, you should probably add country fixed effects.
• If you think that $$Y_{c,y,q}$$ might differ according to the year $$y$$ in addition to what is explained by $$S_{y,q}$$ and $$X_{c,y,q}$$ then you should add year fixed effects.
• If you think that $$Y_{c,y,q}$$ might change with $$q$$ (e.g. seasonality) in addition to $$S_{y,q}$$ and $$X_{c,y,q}$$ then you can add quarter fixed effects.
• In principle you coud add country$$\times$$year fixed effects or country$$\times$$quarter fixed effects if you are worried about a combination of these problems. If you add year$$\times$$quarter fixed effects you loose identification for $$S_{y,q}$$.

Which of these options is best will depend on what exactly you are looking at (i.e. what is $$Y_{c,y,q}$$) and how you think this changes by factors not included in your regression.

If you decide to add year fe, then the identification of the coefficient on the shock $$S_{y,q}$$ (and the interaction) only relies on intra-year variation in $$S_{y,q}$$ and $$Y_{c,y,q}$$. So if you think there is a lot of yearly variation in $$Y_{c,t,q}$$ explained by $$S_{y,q}$$ then you will not be able to capture those when using year fixed effects.

There is also the (subtle) issue of how you define the year. For example, you could define a year from "Januari year $$x$$" to "December year $$x$$". An alternative could be, for example, to define the year from "April year $$x$$" to "March year $$x +1$$". Depending on this choice, your results might vary.

• Thank you very much for your really helpful answer tdm! If you don't mind, I'd ask one follow-up: you didn't touch on this, but my main worry is that adding year fixed effects may introduce inconsistency into my estimates for the coefficient of S and the S x X interaction. I don't have a formal argument but adding fixed effects here feels analogous to adding year fixed effects in a simple 1 country quarterly time series, which (again) feels suspicious. Would part of the common 'global' yearly effect that the year fixed effect subtracts be part of the effect of S as it affects quarters 1,2,..? – lippi May 5 at 11:05
• Suppose I assume that S and X are completely exogenous and my motivation for adding year fixed effects is to soak up some variation from Y and increase the precision of estimates. Thank you very much for you help! – lippi May 5 at 11:06
• If the true model doesn't has fixed effects. Then adding fixed effects does not make the estimates inconsistent, although the estimates will be less efficient in the sense of having larger asymptotic variance (this is the whole argument of the Hausman test for FE versus RE). – tdm May 5 at 11:13
• To add a bit more explanation about my worry: suppose there is a big US shock that makes it a good year for every country in the sample, not only for the 1st quarter but for all subsequent quarters. That means that yearly average Y is high for all countries, which means that I subtract a large number with year fixed effects. This relationship feels strange, but it may just be due to some bad intuition about FE – lippi May 5 at 11:13
• Hmm okay. Thank you very much then. That's really helpful – lippi May 5 at 11:14