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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!

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  • $\begingroup$ 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 $\endgroup$ Apr 9 '20 at 16:40
  • $\begingroup$ 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... $\endgroup$
    – Max
    Apr 9 '20 at 16:55
  • $\begingroup$ what paper what page? $\endgroup$ Apr 9 '20 at 20:00
  • $\begingroup$ Working paper "The Importance of Sovereign Reference Rates for Corporate Debt Issuance: Mind the Gap" p. 15. Appreciate your help! $\endgroup$
    – Max
    Apr 9 '20 at 20:22
  • $\begingroup$ 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? $\endgroup$ Apr 10 '20 at 2:58
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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.

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