History of the fixed effects estimator.

[John Mullahy]

This paper (ungated) may be of interest to many in the Statalist community.
https://www.aeaweb.org/articles?id=10.1257/jep.20241406

[Andrew Musau]

Another important paper in the history of fixed effects is Hoch (Econometrica, 1962). The fixed effects model takes the form:

$$y_{it}= \alpha + \beta x_{it} + \eta_i + u_{it}$$

where \(y_{it}\) is the log of output on farm \(i\) in year \(t\), and \(x_{it}\) is the log of an input on farm \(i\) in year \(t\). In Mundlak (Journal of Farm Economics, 1961), the input is labor, whereas in Hoch, it is fertilizer. The term \(\eta_i\) represents fixed effects—interpreted as management ability in Mundlak (assumed to be correlated with labor input) and as soil quality in Hoch (assumed to be correlated with fertilizer use).

References:
Mundlak, Yair. "Empirical production function free of management bias." Journal of Farm Economics 43, no. 1 (1961): 44-56.
Hoch, Irving. "Estimation of production function parameters combining time-series and cross-section data." Econometrica (1962): 34-53.

[Clyde Schechter]

What a nice article! Thanks for posting that.

[Mukesh Punia]

This paper (ungated) may be of interest to many in the Statalist community.
https://www.aeaweb.org/articles?id=10.1257/jep.20241406

“In two-way fixed effects, unit and time fixed effects are used to account for time-invariant heterogeneity within each unit and unit-invariant heterogeneity within each time period, and thus to get at the causal effect of treatment.” P -270

May I request someone to elaborate little more on this in practical sense.


Andrew Musau that is in reference of #1

Thank you - Mukesh

[Andrew Musau]

This is covered in panel data econometrics textbooks. The two-way FE model is:

\[
y_{it} = \beta x_{it} + \eta_i + \mu_t + u_{it}
\]


where \(\eta_i\) is the unit (e.g., country) fixed effect, \(\mu_t\) is the time fixed effect. So we start with the mechanics of two-way demeaning transformation (or double demeaning):

\[
\tilde{y}_{it} = y_{it} - \bar{y}_i - \bar{y}_t + \bar{y}
\]
\[
\tilde{x}_{it} = x_{it} - \bar{x}_i - \bar{x}_t + \bar{x}
\]


where \(\bar{y}_i\) is the average over time for unit \(i\), \(\bar{y}_t\) is the average over units at time \(t\) and \(\bar{y}\) is the overall average. Applying the transformation to the original model:

\[
y_{it} - \bar{y}_i - \bar{y}_t + \bar{y}
= \beta(x_{it} - \bar{x}_i - \bar{x}_t + \bar{x})
+ (\eta_i - \bar{\eta}_i - \bar{\eta}_t + \bar{\eta})
+ (\mu_t - \bar{\mu}_i - \bar{\mu}_t + \bar{\mu})
+ (u_{it} - \bar{u}_i - \bar{u}_t + \bar{u})
\]

But because \(\bar{\eta}_i = \eta_i\) (since \(\eta_i\) is constant over time), \(\bar{\eta}_t = \frac{1}{N} \sum_i \eta_i = \bar{\eta}\), we have:

\[ \eta_i - \bar{\eta}_i - \bar{\eta}_t + \bar{\eta} = 0\] and;

\[\mu_t - \bar{\mu}_i - \bar{\mu}_t + \bar{\mu} = 0\].


Therefore, the fixed effects cancel out, leaving:

\[ \tilde{y}_{it} = \beta \tilde{x}_{it} + \tilde{u}_{it} \]


This is the within estimator (TWFE), where both unit and time effects have been removed, and \(\beta\) is estimated from the remaining variation in \(x_{it}\) and \(y_{it}\). Now, the assumption is that the heterogeneity is time-invariant or unit-invariant, and the TWFE gives us consistent estimates if this is the case. Stated differently, if you're interested in estimating the causal effect of a policy on some outcome variable, you want to make sure your estimate of \(\beta\) isn't biased by:

- Time-invariant, unit-specific traits, e.g., culture, geography, a country's legal system (removed by \(\eta_i\));

- Unit-invariant, time-specific shocks, e.g., oil prices, pandemics, global recessions (removed by \(\mu_t\)).

By removing both sources of omitted variable bias, two-way fixed effects models isolate the within-unit, over-time variation in \(x_{it}\) that is independent of these unobserved factors.

[Erik Ruzek]

I'm curious how folks here think about some of the recent critiques of the two-way fixed effects models by Kropko & Kubinec (2020) and Imai & Kim (2021).