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A common approach to controlling for unobserved heterogeneity is to run a linear regression with fixed effects. This post shows that this strategy may lead to strong bias in more realistic settings, even if you specify the fixed effects correctly. It is also about some things you can do about it.
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Introduction: the Problem
The Setting
Suppose that we want to estimate the average effect of some variables
\(\boldsymbol{x}\) on some outcome \(y\). For example, \(\boldsymbol{x}\) can
be investment in R&D by a firm, and \(y\) can be worker productivity. In
another example, \(\boldsymbol{x}\) can be air pollution on a given day,
and \(y\) can be the crime rate on that day.
By now everyone agrees that you have to do something about unobserved
heterogeneity — there is a lot of unobserved differences between units
in any economic and business data. Firms differ in their technology,
culture, and managerial capacity, etc. Consumers differ in their
preferences, social ties, and expectations; etc.
Common approach: adding fixed effects A common solution is to obtain
panel data and run a linear regression with “fixed effects”
(FEs):
\(\hspace{1cm} y_{it} = \alpha_i + (\text{other fixed effects}_{it}) + \boldsymbol{\beta}' \boldsymbol{x}_{it} + u_{it}, \quad i=1,\dots, N, \quad t=1, \dots, T \qquad (1)\)
The fixed effects are just heterogeneous constants.
The simplest kind are the unit-level effects \(\alpha_i\). Other effects
can include time-specific intercepts \(\delta_t\) (two-way FEs), or some
more complex combinations, such as industry \(\times\) year interacted
FEs.
The hope is that you will “control” for unobserved heterogeneity if you add “enough” FEs. In this case, \(\boldsymbol{\beta}\) should capture the
effect of \(\boldsymbol{x}\) on \(y\). \(\boldsymbol{\beta}\) can be estimated
by eliminating the FEs from
Equation 1 and
estimating the resulting equation — an approach known as FE (or
within) estimation.
The Problem with Fixed Effects and This Post
But why homogeneous \(\boldsymbol{\beta}\)? A key problem with
Equation 1 is that it
is not an internally consistent model. Why should the slopes
\(\boldsymbol{\beta}\) be the same for all units? This assumption goes
against acknowledging heterogeneity and including FEs. Would it not be
reasonable to specify the more flexible model where the slopes can also
be heterogeneous:
\(\hspace{1cm} y_{it} = \alpha_i + (\text{other fixed effects}_{it}) + \color{red}{\boldsymbol{\beta}_i}' \boldsymbol{x}_{it} + u_{it}, \quad i=1,\dots, N, \quad t=1, \dots, T
\qquad(2)\)
I personally struggle to think of a single convincing example where the
\(\boldsymbol{\beta}\) are homogeneous, while the constant terms can be
heterogeneous.
What is this post about? Problems with FEs and Solutions In this
post, I show that the strategy of using FEs breaks down under the more
realistic Equation 2
and discuss some solutions. Specifically, we will see that:
-
The FE estimator may be catastrophically biased even if you
correctly specify the FEs.
-
If you add additional fixed effects, you may increase bias in
estimation and get worse results than if you didn’t do anything
about heterogeneity.
-
There is a precise condition on the data under which the FE
estimator does correctly estimate the average effects.
-
Some solutions are available: ranging from simply explicitly
thinking about the assumptions needed for consistent estimation to
using estimators robust to parameter heterogeneity.
The full simulation code is available on the blog codes
repository.
The issues with the fixed effects estimators are well-known
at least in the case of two-way effects in diff-in-diff settings (De
Chaisemartin and D’Haultfœuille 2020).
This post argues that the problems with FE estimators are much more
general: the appear in all contexts, no just diff-in-diff, and affect
any configuration of fixed effects.
A Simulation Study: the Issues of the FE Estimator
Setting and Estimators Compared
Let us start with the problems. It is easiest to show the issue with a
small simulation study.
Outcome equation Suppose that the outcome \(y_{it}\) is generated with
a very simple model which only has one individual fixed effect
\(\alpha_i\) and only two periods of data:
\[y_{it} = \alpha_i + (\beta + \alpha_i) x_{it} + u_{it}, \quad i=1, \dots, N, \quad t=1, 2, \quad \mathbb{E}[u_{it}| x_{i1}, x_{i2}] =0,\]
where \(\beta\) is the parameter of interest — the average coefficient
vector (average effect). We set \(\beta\) to be negative:
\[\beta = - 0.25.\]
There are only two types of units in terms \(\alpha_i\): \(\alpha_i\) is
equal to \(+1\) and \(-1\) with equal probabilities. Accordingly, half the
units in population have a positive slope of \(0.75\), and half have a
negative slope of \(-1.25\).
Estimators compared: FE vs. no FE We compare two estimators for
\(\beta\):
-
Simple pooled OLS regression (no fixed effects), which involves
estimating the model
\[y_{it} = \beta x_{it} + w_{it}, \quad\quad\quad w_{it} = \alpha_i + \alpha_i x_{it} + u_{it}\]
-
The fixed-effects/within estimator that correctly specifies the
unit-level random intercepts. In this case we estimate the following
model:
\[y_{it} = \alpha_i + \beta x_{it} + v_{it}, \quad\quad\quad v_{it} = \alpha_i x_{it} + u_{it}.\]
While just these two estimators are sufficient to illustrate the point,
it is straightforward to consider designs with more random intercepts —
time effects, interacted effects, different groupings, etc.
The rest of the DGP The rest of the data-generating process (DGPs)
is such that the pooled OLS estimator is unbiased, while the FE
estimator is biased. See the fold-out for details.
Details of the GDP
The rest of the DGP is specified as follows:
-
Conditional on a given value of \(\alpha_i, (x_{i1}, x_{i2})\) is
bivariate normal with parameters that depend on whether
\(\alpha_i = +1\) or \(\alpha_i = -1\):
\[\begin{pmatrix} x_{i1} \\ x_{i2} \end{pmatrix}\Bigg| \alpha_i = + 1 \sim N\left( \begin{pmatrix} \mu_{1+} \\ \mu_{2+} \end{pmatrix}, \begin{pmatrix} \sigma_{1+}^2 & \rho_+ \sigma_{1+} \sigma_{2+} \\ \rho_+ \sigma_{1+} \sigma_{2+} & \sigma_{2+}^2 \end{pmatrix} \right)\]
and
\[\begin{pmatrix} x_{i1} \\ x_{i2} \end{pmatrix}\Bigg| \alpha_i = - 1 \sim N\left( \begin{pmatrix} \mu_{1-} \\ \mu_{2-} \end{pmatrix}, \begin{pmatrix} \sigma_{1-}^2 & \rho_- \sigma_{1-} \sigma_{2-} \\ \rho_- \sigma_{1-} \sigma_{2-} & \sigma_{2-}^2 \end{pmatrix} \right).\]
-
\(u_{it}\) is standard normal and independent of the other components
in the model.
We set the parameters for the distribution of \(x_{it}\) so that the OLS
estimator is consistent and the FE estimator is inconsistent.
Specifically, we select the parameters so that the following conditions
are satisfied:
\[\begin{aligned}
\mathbb{E}[x_{it}(\alpha_i + \alpha_i x_{it}+ u_{it})] & = 0, \quad t=1, 2, \\
\mathbb{E}[(x_{i2}- x_{i1})(\alpha_i (x_{i2}- x_{i1})+ u_{it})] & \neq 0 .
\end{aligned}\]
Note that \((x_{i2}-x_{i1})\) is exactly the within-transformed
regressor involved in the FE regression.
We find some parameters by simply specifying a nonzero value for the
second expectations and searching for a root of the resulting equations — simply applying GMM in population.
Here is a simple Python implementation of a GMM solver class:
import numpy as np
from scipy.optimize import minimize
from typing import Callable, List, Dict, Any, Optional
class GMMSolver:
"""
A Generalized Method of Moments (GMM) solver that estimates parameters by
minimizing the weighted squared distance of moment conditions from zero.
Attributes:
moment_conditions (Callable[[np.ndarray], np.ndarray]):
Function returning moment conditions given parameter values.
constraints (Optional[List[Dict[str, Any]]]):
List of constraints for parameter optimization.
initial_guess (np.ndarray):
Initial parameter values for the optimization.
weighting_matrix (np.ndarray):
Weighting matrix for the GMM objective function. Defaults to identity.
process_func (Callable[[np.ndarray], Dict[str, Any]]):
Function that processes the optimized parameters into a meaningful format.
"""
def __init__(
self,
moment_conditions: Callable[[np.ndarray], np.ndarray],
initial_guess: np.ndarray,
constraints: Optional[List[Dict[str, Any]]] = None,
weighting_matrix: Optional[np.ndarray] = None,
process_func: Optional[Callable[[np.ndarray], Dict[str, Any]]] = None
) -> None:
"""
Initializes the GMM solver with the moment conditions and optimization settings.
Args:
moment_conditions (Callable[[np.ndarray], np.ndarray]):
A function that takes a parameter vector and returns moment conditions.
initial_guess (np.ndarray):
Initial parameter values for optimization.
constraints (Optional[List[Dict[str, Any]]], optional):
Constraints for optimization. Defaults to None.
weighting_matrix (Optional[np.ndarray], optional):
Weighting matrix for GMM. Defaults to identity matrix.
process_func (Optional[Callable[[np.ndarray], Dict[str, Any]]], optional):
Function to format the final parameter estimates. Defaults to None.
"""
self.moment_conditions = moment_conditions
self.initial_guess = np.array(initial_guess)
self.constraints = constraints if constraints else []
self.weighting_matrix = (
np.eye(len(moment_conditions(initial_guess))) if weighting_matrix is None else weighting_matrix
)
self.process_func = process_func
self.estimated_params = None
def _gmm_objective(self, params: np.ndarray) -> float:
"""
Computes the GMM objective function:
m(θ)ᵀ W m(θ), where m(θ) are the moment conditions.
Args:
params (np.ndarray): Current parameter estimates.
Returns:
float: The GMM loss function value.
"""
moments = self.moment_conditions(params)
return moments.T @ self.weighting_matrix @ moments
def minimize(self) -> None:
"""
Runs the GMM estimation by minimizing the GMM objective function.
Returns:
np.ndarray: The estimated parameters.
"""
result = minimize(self._gmm_objective, self.initial_guess, constraints=self.constraints)
if not result.success:
raise ValueError(f"Optimization failed: {result.message}")
self.estimated_params = result.x
def process_solution(self) -> Dict[str, Any]:
"""
Processes the estimated parameters into a meaningful format.
Returns:
Dict[str, Any]: Processed output, depends on user-supplied processing function.
"""
if self.process_func is None:
return {"parameters": self.estimated_params}
return self.process_func(self.estimated_params)
The moments, constraints, and processing functions are defined as follows
def sim_moment_conditions(params: np.ndarray) -> np.ndarray:
"""
Moment conditions for consistency of OLS and inconsistency of FE estimators.
Args:
params (np.ndarray): Parameter values.
Returns:
np.ndarray: Moment equations evaluated at given parameters.
"""
sigma1p, sigma2p, rhop, mu1p, mu2p, sigma1m, sigma2m, rhom, mu1m, mu2m = params
return np.array([
sigma2p**2 + mu2p**2 + mu2p - sigma2m**2 - mu2m**2 - mu2m,
sigma1p**2 + mu1p**2 + mu1p - sigma1m**2 - mu1m**2 - mu1m,
sigma1p**2 + sigma2p**2 - 2 * rhop * sigma1p * sigma2p
+ mu2p**2 + mu1p**2 - 2 * mu1p * mu2p
- sigma1m**2 - sigma2m**2 + 2 * rhom * sigma1m * sigma2m
- mu2m**2 - mu1m**2 + 2 * mu1m * mu2m
- 1000
])
def process_mu_sigma_params(params: np.ndarray) -> dict:
"""
Extracts mu and sigma matrices from the parameters vector.
Args:
params (np.ndarray): Estimated parameter values.
Returns:
dict: Processed parameters (mu and sigma matrices).
"""
mu_plus = params[3:5]
mu_minus = params[8:]
sigma_plus = np.array([
[params[0]**2, (params[0] * params[1]) * params[2]],
[(params[0] * params[1]) * params[2], params[1]**2]
])
sigma_minus = np.array([
[params[5]**2, (params[5] * params[6]) * params[7]],
[(params[5] * params[6]) * params[7], params[6]**2]
])
return {"mu_plus": mu_plus,
"mu_minus": mu_minus,
"sigma_plus": sigma_plus,
"sigma_minus": sigma_minus,
}
# Constraints on data-generating process parameters
constraints = [
{'type': 'ineq', 'fun': lambda vars: vars[0]}, # sigma_{1+} >= 0
{'type': 'ineq', 'fun': lambda vars: vars[1]}, # sigma_{2+} >= 0
{'type': 'ineq', 'fun': lambda vars: 1 - vars[2]}, # rho_+ <= 1
{'type': 'ineq', 'fun': lambda vars: vars[2] + 1}, # rho_+ >= -1
{'type': 'ineq', 'fun': lambda vars: vars[5]}, # sigma_{1-} >= 0
{'type': 'ineq', 'fun': lambda vars: vars[6]}, # sigma_{2-} >= 0
{'type': 'ineq', 'fun': lambda vars: 1 - vars[7]}, # rho_- <= 1
{'type': 'ineq', 'fun': lambda vars: vars[7] + 1}, # rho_- >= -1
]
# Initial guess for parameters
param_initial_guess = [12, 15, 0.3, 24, -7, 2, 8, 0.6, 5, 14]
Finally, we can compute the parameters as:
solver_dgp_params = GMMSolver(
sim_moment_conditions,
param_initial_guess,
constraints,
process_func=process_mu_sigma_params,
)
solver_dgp_params.minimize()
mu_sigma_params = solver_dgp_params.process_solution()
print(mu_sigma_params)
{'mu_plus': array([ 1.09064557, -22.60362844]),
'mu_minus': array([13.7704972 , 25.70151531]),
'sigma_plus': array([[437.04406675, 64.35128113],
[ 64.35128113, 440.05540533]]),
'sigma_minus': array([[235.927077 , 155.28354406],
[155.28354406, 242.10637242]])}
The values you obtain may differ — there are multiple sets of parameters
that yield the desired values for the expectations.
Unfortunately, such a DGP is as plausible as an opposite one where only
the FE estimator is consistent. Intuitively, the two cases are equally
likely because both involve the same number of restrictions on the
parameters (though in this sense both DGPs are less likely than the
situation in which both estimators are inconsistent).
As a consequence, one should not dismiss this DGP as an unrealistic edge
case.
Results: Distribution of Estimators
Distribution of estimators Let us take a look at the distribution of
the two estimators. The following pictures shows the densities of the
estimators for a few values of \(N\). I also highlighted the target value
— the average effect \(\beta=0.25\).
Pooled OLS (no fixed effects) does pretty well. It is unbiased and
becomes fairly precise even in pretty small sample sizes.
However, adding fixed effects dramatically worsens the situation. The FE
estimator is strongly biased. The bias is so large that the target of
the FE estimator (a) has a different sign from \(\beta\) and (b) is larger
(in absolute value) than the target effect.
Wrong effect sign with FE As a consequence, if you trust the FE
estimator, you will with probability \(\approx 1\) conclude that there is
a positive effect on average, even for tiny sample sizes:
Takeaway: Do Not Trust Estimate Changes after Adding Fixed Effects
So what does this simulation tell us?
First, the FE estimator can be significantly biased even if the fixed
effects themselves are correctly specified. The bias may be so large
that not even the sign of the average effect is correctly estimated.
Second, adding FEs may worse estimation quality. The simple example
above shows how a smaller model yields estimates preferable to those of
a larger model. One should be careful with the false “intuition” that
adding more FE better controls for heterogeneity.
When the FE Estimators Is Usable (and When Not)
Is there anything that can be done to estimate the average effect under
Equation 2? Several
things, as we discuss below. As the first answer, we will consider the
conditions under which the FE can yield a consistent estimate for the
average effect.
Conditions for Consistency of the FE Estimator
What controls whether the fixed effects estimator is estimating the
average effect? Or, put differently, what configurations of fixed
effects permit consistent estimation of the average effect with the FE
estimator?
Theoretical framework It is straightforward to work out the exact
conditions in a fairly general case. Suppose that \(y_{it}\) is generated as
\[y_{it} = \text{fixed effects}_{it} + \boldsymbol{\beta}_i' \boldsymbol{x}_{it} + u_{it}, \quad i=1,\dots, N, \quad t=1, \dots, T, \quad \mathbb{E}[u_{it}|\boldsymbol{x}_{i1}, \dots, \boldsymbol{x}_{iT}]=0,\]
where \(\boldsymbol{x}_{it}\) is a vector of covariates. Here
\(\text{fixed effects}_{it}\) can stand for one-, two-, or multi-way fixed
effects. Any configuration of fixed effects is allowed, as long as it
can be eliminated by a suitable within transformation without deleting
the full data completely. Let us assume that the residual \(u_{it}\) is
strictly exogenous in the sense that
\(\mathbb{E}[u_{it}|\boldsymbol{x}_{i1}, \dots, \boldsymbol{x}_{iT}]=0\).
This condition rules out dynamic panels where the situation is more
complex and the subject of a future post.
Formal definition of the FE estimator Apply the within
transformation to eliminate the fixed effects. Label the resulting
variables \(\tilde{w}_{it}\) so that the within-transformed variables
satisfy
\[\tilde{y}_{it} = \boldsymbol{\beta}_i'\tilde{\boldsymbol{x}}_{it} + \tilde{u}_{it}.\]
For each unit \(i\), stack \(\tilde{\boldsymbol{x}}_{it}\) into the
matrix \(\tilde{\boldsymbol{X}}_i\). Similarly combine all
\(\tilde{y}_{it}\) for a given unit $i$ into the vector
\(\tilde{\boldsymbol{y}}_i\). The fixed effect estimator
\(\hat{\boldsymbol{\beta}}_{FE}\) can be written as
\[\hat{\boldsymbol{\beta}}^{FE} = \left(\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{y}}_i.\]
For convenience, define \(\boldsymbol{\eta}_i\) as
\[\boldsymbol{\eta}_i = \boldsymbol{\beta}_i - \mathbb{E}[\boldsymbol{\beta}_i] .\]
Characterization of consistency of the FE estimator The following
theorem shows that \(\hat{\boldsymbol{\beta}}^{FE}\) consistently
estimates \(\mathbb{E}[\boldsymbol{\beta}_i]\) if and only if a particular
orthogonality condition holds between the second moments of
\(\tilde{\boldsymbol{X}}_i\) and the deviations \(\boldsymbol{\eta}_i\) of
the coefficients from their mean.
RESULT: \(\hat{\boldsymbol{\beta}}^{FE}\) converges to
\(\mathbb{E}[\boldsymbol{\beta}_i]\) in probability as \(N\to\infty\) if
and only if
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i] =0.\)
This result first appears in Wooldridge (2003) and Wooldridge (2005).
Proof
Throughout, we assume that all corresponding moments exists and all
sample average converge to population averages.
We first expand the FE estimator using the within-transformed model:
$$
\begin{aligned}
\hat{\boldsymbol{\beta}}^{FE} & = \left(\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{y}}_i \\
& = \left(\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{X}}_i\boldsymbol{\beta}_i + \left(\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{u}}_i.
\end{aligned}
$$
Substituting the definition of the deviations from the average coefficients nets us
$$
\begin{aligned}
\hat{\boldsymbol{\beta}} & = \mathbb{E}[\boldsymbol{\beta}_i] + \left( \dfrac{1}{N}\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \dfrac{1}{N}\sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i + \left( \dfrac{1}{N}\sum_{i=1}^N \tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right)^{-1} \dfrac{1}{N}\sum_{i=1}^N \tilde{\boldsymbol{X}}_i \tilde{\boldsymbol{u}}_i\\
& \xrightarrow{p} \mathbb{E}[\boldsymbol{\beta}_i] + \left(\mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right] \right)^{-1} \mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i \right] + \left(\mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right] \right)^{-1} \mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{u}_i \right]\\
& = \mathbb{E}[\boldsymbol{\beta}_i] + \left(\mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i \right] \right)^{-1} \mathbb{E}\left[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i \right].
\end{aligned}
$$
The result can now be read off from the last display.
A Sufficient Condition for \(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]=0\)
It is not obvious when
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]=0\).
To me, it is usually not even clear what it means — the condition
involves \(\boldsymbol{\eta}_i\) and the second moments of the
within-transformed covariates.
Strong sufficient condition: full independence The condition will
certainly hold if \(\boldsymbol{\beta}_i\) and \(\boldsymbol{X}_i\) are
independent. However, you only really expect this independence in
marginally randomized experiments. It is extremely unlikely to hold in
observational data.
More general sufficient condition: mean independence Luckily, there
is a simple sufficient condition that does not require full
independence. It is sufficient that \(\boldsymbol{\eta}_i\) is
mean-independent from \(\tilde{\boldsymbol{X}}\) in the following sense:
\[\mathbb{E}[\boldsymbol{\eta}_i|\tilde{\boldsymbol{X}}_i] = 0.\]
Intuition: case of unit-level FEs To get some intuition, consider
the simplest case where only individual-level effects are present:
\[y_{it} = \alpha_i + \boldsymbol{\beta}_i'\boldsymbol{x}_{it} + u_{it}.\]
In this case, the within transformation simply subtracts individual
time averages, that is,
\[\tilde{\boldsymbol{x}}_{it} = \boldsymbol{x}_{it} - \dfrac{1}{T} \sum_{t=1}^T \boldsymbol{x}_{it}.\]
The within transformation removes permanent (=static) components in
\(\boldsymbol{x}_{it}\). What is left are the dynamics in
\(\boldsymbol{x}_{it}\). Accordingly, the above condition requires that
the changes in \(\boldsymbol{x}_{it}\) over time are not informative about
the individual coefficients.
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]\) and Adding More Effects
Can we say anything about how
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]\)
or \(\mathbb{E}[\boldsymbol{\eta}_i|\tilde{\boldsymbol{X}}_i]\) as we add
more fixed effects?
Unfortunately, very little can be said in general. Adding more FEs
changes the definition of \(\tilde{\boldsymbol{X}}_i\) and reduces the
remaining variation in the covariates. In principle, this reduction
makes it easier for the condition
\(\mathbb{E}[\boldsymbol{\eta}_i|\tilde{\boldsymbol{X}}_i]=0\) to be
satisfied, since the expectation conditions on progressively less
information. In the limit the fixed effects are rich enough so that
\(\tilde{\boldsymbol{X}}_i=\varnothing\), and then automatically
\(\mathbb{E}[\boldsymbol{\eta}_i|\tilde{\boldsymbol{X}}_i]=\mathbb{E}[\boldsymbol{\eta}_i]=0\).
However, at this point there is no variation left in the covariate, so
no estimation is possible.
It is also generally impossible to predict the direction of change in
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]\)
as an additional layer of FEs is added. Such an addition may either
bring
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]\)
closer to zero, or push it further away, as in the simulation.
Practical Advice and Conclusions
So what should one do in practice? Ignoring the issue seems unreasonable
and internally inconsistent, as pointed out in the introduction.
However, there are (at least) three solutions which seem reasonable to me.
Solution 1: An Honest Approach to the FE Estimator
The minimal solution is to at least acknowledge that the FE estimator is
only consistent under the condition that
\(\mathbb{E}[\tilde{\boldsymbol{X}}_i'\tilde{\boldsymbol{X}}_i\boldsymbol{\eta}_i]=0\)
(or some other version of this condition appropriate to the assumed pattern of heterogeneity in the coefficients). If you wish to trust the FE estimator, you
should first justify why the above condition holds in your particular
empirical setting.
Solution 2: A Robust Approach with the Mean Group Estimator
A more ambitious solution is to use a heterogeneity-robust approach —
the mean group estimator (Pesaran and Smith 1995) or one of its
refinements (Breitung and Salish 2021). This approach has a data price —
\(T\) must be larger than the number of covariates. If this condition is
satisfied and there is enough variation in the individual data, one can
then run unit-by-unit regressions. The mean group estimator is simply
the cross-sectional average of these individual estimators. It is not
hard to show that it is consistent for
\(\mathbb{E}[\boldsymbol{\beta}_i]\) without any assumptions on how
\(\boldsymbol{\beta}_i\) and \(\boldsymbol{X}_i\) are related.
Solution 3: A Robust Approach with Grouped Estimation
A different flexible approach is to try grouped estimators in the style
of Bonhomme and Manresa (2015). This approach is more appropriate if \(T\)
is large and one is unwilling to believe that the coefficients \(\beta_i\)
are invariant in the data.
Conclusions
In summary, one should not consider the strategy of specifying fixed
effects as a sufficient way of controlling for heterogeneity. Under
parameter heterogeneity, the FE estimator may in general be quite biased even if the FEs are specified correctly. To solve this issue, one should either think carefully about the empirical setting or use
heterogeneity-robust approaches.
References
- Bonhomme, Stéphane, and Elena Manresa. 2015.
“Grouped Patterns of Heterogeneity in Panel
Data.” Econometrica 83 (3): 1147–84. https://doi.org/10.3982/ecta11319.
- Breitung, Jörg, and Nazarii Salish. 2021.
“Estimation of Heterogeneous Panels with Systematic
Slope Variations.” Journal of Econometrics 220 (2): 399–415.https://doi.org/10.1016/j.jeconom.2020.04.007.
- De Chaisemartin, Clément, and Xavier D’Haultfœuille. 2020. “Two-Way
Fixed Effects Estimators with Heterogeneous Treatment Effects.”
American Economic Review 110 (9): 2964–96. https://doi.org/10.1257/aer.20181169.
- Pesaran, M. Hashem, and Ron P. Smith. 1995.
“Estimating long-run relationships from dynamic
heterogeneous panels.” Journal of Econometrics 6061: 473–77.
- Wooldridge, Jeffrey M. 2003. “Fixed Effects
Estimation of the Population-Averaged Slopes in a Panel Data Random
Coefficient Model.” Econometric Theory 19: 411–13. https://doi.org/10+10170S0266466603002081.
- ———. 2005. “Fixed-effects and related estimators
for correlated random-coefficient and treatment-effect panel data
models.” The Review of Economics and Statistics 87 (May):
385–90.
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