# Different fixed effect models

I am stuck at understanding the difference between different types of fixed effect models. In particular, what is the difference between:

1. A two way fixed effect model in the form of $$y_{it} = \alpha_i + \gamma_t + \mathbf x_{it}^\top \beta + \epsilon_{it},$$ where $$\gamma_t$$ is the time fixed effect and $$\alpha_i$$ the entity fixed effect. In the literature this is called two-way fixed effect most of the time.

2. A fixed effect model in the type of

$$y_{it} = \sigma_{it} + \mathbf x_{it}^\top \beta + \epsilon_{it},$$ where the time fixed effects and the entity fixed effects are combined in one estimator. That would mean you would have a time-entity dummy for each combination. Are there any paper regarding this kind of fixed effect model?

Some explanation would be really nice.

Thank you!

• The terms fixed and random are used by different people (or the same people) at different times to mean different things. See: statmodeling.stat.columbia.edu/2005/01/25/why_i_dont_use Apr 9, 2020 at 16:40
• So, in the second case, when combining entity and time fixed effects, it is rather a random effects model? My question results from an excerpt in a paper, where it says: Furthermore, country x 5-year fixed effects capture general, time-varying institutional and policy factors that can affect the shape of the yield curve...
– Max
Apr 9, 2020 at 16:55
• what paper what page? Apr 9, 2020 at 20:00
• Working paper "The Importance of Sovereign Reference Rates for Corporate Debt Issuance: Mind the Gap" p. 15. Appreciate your help!
– Max
Apr 9, 2020 at 20:22
• Welcome Max! This is a quote from page 22 of the aforementioned working paper. It appears they are referring to the interaction between country and year. Is your question specific to a single ‘entity-year’ fixed effect, and not the interaction between two separate fixed effects for entity and year? Apr 10, 2020 at 2:58

The two way fixed effect model (TWFE)

$$y_{it} = \alpha_i + \gamma_t + \mathbf x_{it}^\top \beta + \epsilon_{it},$$

is treated in the seminal paper Abowd, Kramerz and Magolis (1999) High Wage Workers and High Wage Firms.

The model

$$y_{it} = \sigma_{it} + \mathbf x_{it}^\top \beta + \epsilon_{it},$$

cannot be estimated using within estimator under the standard fixed effect assumption that $$\sigma_{it}$$ is correlated with $$\mathbf x_{it}$$ because there is a fixed effect for each observation $$(i,t)$$ (this is completely analogously to allowing for individual fixed effects with only cross-section data $$y_{i} = \mu_{i} + \mathbf x_{i}^\top \beta + \epsilon_{i}$$).

Still in comparing the two models it is evident that the first model can be achieved by the restricting assumption that $$\sigma_{it} = \alpha_i + \delta_t$$. Restricting because the second model allows for individual specific time path $$\sigma_{i,t_1},\sigma_{i,t_2},...,\sigma_{i,T}$$ whereas the first model assumes the timepath $$\delta_{t_1},\delta_{t_2},...,\delta_{T}$$ is the same for all individuals.

In the working paper you cite the authors have data from 14 countries in the period 1991-2017 and they focus on what they refer to as "country-year-maturity bins".

Let the index $$c=1,...,14$$ be the country index and $$t=1,...,27$$ be the year index. Then for each country $$c$$ at time $$t$$ they have observations on a set of maturity bins $$k=1,...,9$$. Their number of observations should therefore be $$C*T*K = 14*27*9$$. The model may be written as

$$y_{kct} = \sigma_{ct} + x_{kct}^\top \beta + \epsilon_{kct}$$

so the fact that there are several maturity bins $$k=1,..,K$$ allows application of the within estimator. And the fact that the fixed effects for country $$\alpha_c$$ and time $$\delta_t$$ are interacted allows for a more general model where country specific timepaths are allowed.