# How should I interpret a Two-way-fixed-effects-regression?

I have problems with the interpretation of a Two-way-fixed-effects-model. I understood both the interpretation of the single individual effect and the single time effect in a fixed-effects-model. But how can I interpret the model if both effects are included in the model?

$$y_{ i,t }=x'_{ i,t }\beta+\alpha_{ i }+\theta_{ t }+\epsilon_{ i,t }$$

For example, I have a panel of firms, $$\theta_{ t }$$ might represent business cycle effects, whereas $$\alpha_{ i }$$ would contain firm specific effects that can be argued to be constant over time, such as the "culture" of the firm.

How should I interpret this Two-way-model? Are there business cycle effects for every firm culture? Or do both effects exist without interacting?

## 1 Answer

This approach - including fixed effects for both individual (firms in your case) and for time period, is commonly known as two-way fixed effects. We are looking at variation within firms as well as within year at this point. So we can think of the variation that’s left as being variation relative to what we’d expect given that firm, and given that year.

$$\theta_{t}$$ are time-fixed effect, e.g. shocks that happen every year and hit equally all firms.

$$\alpha_{i}$$ as you said, are firm effects that are constant over time.

The effect of a covariate takes into account both effects. FOR EXAMPLE, let's say the outcome variable ($$Y_{it}$$) is firms revenues, and we are checking the effect of a covariate ($$X_{it}$$) as advertisement investment. We do happen to know that the year 2009 was a particularly bad year for the firms, what with the Great Recession and all. Let’s say that revenues in 2009 were 10,000 below what they normally are. And let’s say that we’re looking at Amazon, who happens to have a revenue of 120,000 per year, a lot more money than the average firm (40,000).

In 2009 Amazon's revenue was 16,000. That’s way above average, but that can be explained by the fact that it’s Amazon and Amazon earns a lot. So given that it’s Amazon, it’s 4,000 below average. But given that it’s 2009, most people are earning 10,000 less, but Amazon is only earning 4,000 less. So given that it’s Amazon, and given that it’s 2009, Amazon in 2009 is 6,000 more than you’d expect.

Reference: Huntington-Klein, Nick. The effect: An introduction to research design and causality. Chapman and Hall/CRC, 2022.