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:


*

*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.

*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!
 A: 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.
