# 2-way Fixed Effects

Instance of:

AKA:

Distinct from:

English:

Formalization:

No fixed effects, single intercept, fully pooled model $y_{it}=\alpha + \beta x_{it} + \epsilon_{it}$

Unit fixed effects $y_{it}=\alpha_i + \beta x_{it} + \epsilon_{it}$

Where $$i$$ represent units.

Time fixed effects $y_{it}=\alpha_t + \beta x_{it} + \epsilon_{it}$ ::: {.column-margin} Where $$t$$ represent time points. :::

2-way fixed effects, both unit and time $y_{it}=\alpha_i+\alpha_t + \beta x_{it} + \epsilon_{it}$

“We demonstrate that the two-way FE model makes specific assumptions about TSCS datasets, and if these assumptions are not met, the model may be unidentified even if substantial variation exists along both dimensions. Because of the difficulty in interpretation, we do not recommend that applied researchers rely on the two-way FE model except for situations in which the assumptions are well-understood, such as the canonical difference-in-difference design.”

“These models can be considered special cases of multilevel or hierarchical models in which the case or time-specific intercepts are assumed to have improper uniform prior distributions, or equivalently normal prior distributions with infinite variance.”

Employing a one-way FE model in a way that can answer the research question, on the other hand, does not guarantee that it will do so. That is, selecting a model with a correct interpretation is a necessary but not a sufficient condition for successful statistical analysis. Indeed, time series in TSCS data may have all of the well known problems of time series in non-panel contexts: seasonality, non-stationarity, stochastic volatility, and so on. Likewise, cross-sections in TSCS data may exhibit reverse causality, heteroskedasticity, multicollinearity, etc. Both time series and cross-sections can also suffer from omitted variable bias if there are unmeasured confounders along the dimension of variance in the model. While these issues must be addressed, they must be addressed in a way that still allows researchers to interpret the model.

• The punchline is that fixed effects are pitched as a magical solution to unmeasured confounders. Magic doesn’t exist, so it must be doing something else. Fixed effects are isolating a portion of observed variation, and estimating a regression model on that smaller transformed domain. Conterfactuals on that new smaller domain are representative of the full real domain of interest only under very specific conditions, which probably don’t hold.

• The coefficients of 2 way fixed effects are nearly uninterpretable and almost surely do not match the research question.

• 2 way fixed effects are oddly closer to the fully pooled model, providing a weighted average of within case and within time effects.

We provide a critical discussion of 12 limitations, including a culture of omission, low statistical power, limited external validity, restricted time periods, measurement error, time invariance, undefined variables, unobserved heterogeneity, erroneous causal inferences, imprecise interpretations of coefficients, imprudent comparisons with cross-sectional models, and questionable contributions vis-à-vis previous work.

use a decomposition of fixed effects models in terms of weights that allows them to re-construct the estimator as a weighted average, permitting non-parametric inference.

we use potential outcomes to define causal quantities of interest in these settings and clarify how standard models like the autoregressive distributed lag model can produce biased estimates of these quantities due to post-treatment conditioning. We then describe two estimation strategies that avoid these post-treatment biases—inverse probability weighting and structural nested mean models

## References

Blackwell, Matthew, and Adam N. Glynn. 2018. “How to Make Causal Inferences with Time-Series Cross-Sectional Data Under Selection on Observables.” American Political Science Review 112 (4): 1067–82. https://doi.org/10.1017/S0003055418000357.
Hill, Terrence D., Andrew P. Davis, J. Micah Roos, and Michael T. French. 2020. “Limitations of Fixed-Effects Models for Panel Data.” Sociological Perspectives 63 (3): 357–69. https://doi.org/10.1177/0731121419863785.
Imai, Kosuke, and In Song Kim. 2019. “When Should We Use Unit Fixed Effects Regression Models for Causal Inference with Longitudinal Data?” American Journal of Political Science 63 (2): 467–90. https://doi.org/10.1111/ajps.12417.
Kropko, Jonathan, and Robert Kubinec. 2020. “Interpretation and Identification of Within-Unit and Cross-Sectional Variation in Panel Data Models.” PLOS ONE 15 (4): e0231349. https://doi.org/10.1371/journal.pone.0231349.