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Political Analysis
--------------------------------------------------------------------------------
ARTICLE CONTENTS
* Abstract
* Introduction
* The Sources of and Potential for Dynamic Misspecification
* The Bias of the Fixed-Effects Estimator with Dynamic Misspecification
* Design of the Monte Carlo Experiments
* Results
* Conclusion
* Footnotes
* References
NOT SO HARMLESS AFTER ALL: THE FIXED-EFFECTS MODEL
Published online by Cambridge University Press: 04 December 2018
Thomas Plümper and
Vera E. Troeger
Show author details
--------------------------------------------------------------------------------
Thomas Plümper Affiliation:
Vienna University of Economics, Department of Socioeconomics, Welthandelsplatz
1, 1020 Vienna, Austria. Email: thomas.pluemper@wu.ac.at
Vera E. Troeger* Affiliation:
University of Warwick, Department of Economics and CAGE, Coventry CV4 7AL, UK.
Email: v.e.troeger@warwick.ac.uk
*
*Email: v.e.troeger@warwick.ac.uk
--------------------------------------------------------------------------------
Article
* Article
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Article contents
* Abstract
* Introduction
* The Sources of and Potential for Dynamic Misspecification
* The Bias of the Fixed-Effects Estimator with Dynamic Misspecification
* Design of the Monte Carlo Experiments
* Results
* Conclusion
* Footnotes
* References
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ABSTRACT
The fixed-effects estimator is biased in the presence of dynamic
misspecification and omitted within variation correlated with one of the
regressors. We argue and demonstrate that fixed-effects estimates can amplify
the bias from dynamic misspecification and that with omitted time-invariant
variables and dynamic misspecifications, the fixed-effects estimator can be more
biased than the ‘naïve’ OLS model. We also demonstrate that the Hausman test
does not reliably identify the least biased estimator when time-invariant and
time-varying omitted variables or dynamic misspecifications exist. Accordingly,
empirical researchers are ill-advised to rely on the Hausman test for model
selection or use the fixed-effects model as default unless they can convincingly
justify the assumption of correctly specified dynamics. Our findings caution
applied researchers to not overlook the potential drawbacks of relying on the
fixed-effects estimator as a default. The results presented here also call upon
methodologists to study the properties of estimators in the presence of multiple
model misspecifications. Our results suggest that scholars ought to devote much
more attention to modeling dynamics appropriately instead of relying on a
default solution before they control for potentially omitted variables with
constant effects using a fixed-effects specification.
--------------------------------------------------------------------------------
KEYWORDS
consistencyefficiencymisspecificationomitted variable biaspanel dataMonte Carlo
simulation
--------------------------------------------------------------------------------
Type Articles
Information
Political Analysis , Volume 27 , Issue 1 , January 2019 , pp. 21 - 45
DOI: https://doi.org/10.1017/pan.2018.17 [Opens in a new window]
Copyright
Copyright © The Author(s) 2018. Published by Cambridge University Press on
behalf of the Society for Political Methodology.
1 INTRODUCTION
The reputation of the fixed-effects estimatorFootnote 1 is better than its
finite sample properties. Among the panel and pooled analysis textbooks that we
are aware of, Wooldridge has perhaps the most precise description of the
conditions under which fixed-effects models are unbiased: “Under a strict
exogeneity assumption on the explanatory variables, the fixed-effects estimator
is unbiased.” (Wooldridge Reference Wooldridge2002, 442) This in turn means that
if the variables included in the model are correlated with a model
misspecification other than omitted variables with constant effects, the
fixed-effects model is not unbiased. For example, recent research has argued
that in the presence of dynamic misspecification, fixed-effects estimates are
biased and inconsistent (Harris et al. Reference Harris, Kostenko, Matyas and
Timol2009; Lee Reference Lee2012; Ahn, Lee, and Schmidt Reference Ahn, Lee and
Schmidt2013; see also Nickell Reference Nickell1981).Footnote 2
We take this literature one step further and demonstrate that fixed-effects
estimates amplify the bias from dynamic misspecificationFootnote 3 relative to
pooled-OLS estimates. This finding has two implications: In the absence of
omitted time-invariant variables and the presence of dynamic misspecification,
the pooled-OLS model is strictly less biased than the fixed-effects model. And
second, in the simultaneous presence of omitted variables with both constant and
time-varying effects, the fixed-effects model is more biased than the pooled-OLS
model (and the random-effects model) if the correlation between the variable of
interest and the omitted time-invariant variance is smaller than the correlation
between the variable of interest and the omitted time-varying variance.
Accordingly, relative to the naïve pooled-OLS benchmark, the fixed-effects model
solves the problem of omitted variables with constant effects at the expense of
rendering other problems worse.Footnote 4
We use the pooled-OLS estimator as a benchmark for the fixed-effects model in
the following sense: The properties of the pooled-OLS estimator in the presence
of omitted time-invariant variables, omitted time-varying variables, and dynamic
misspecifications are known to be poor. Pooled-OLS gives biased estimates in the
presence of omitted time-invariant variables, omitted time-varying variables and
other dynamic misspecifications.
By demonstrating that the fixed-effects model often performs worse than the
pooled-OLS estimator when dynamic misspecifications exist, we try to alert
applied researchers about the importance of choosing the correct dynamic
specification when relying on fixed-effects estimates. Of course, we do not
argue that ignoring dynamics and using the pooled-OLS model is an appropriate
alternative. Optimally, scholars would use the correct dynamic specification for
their model. However, in many applications, the chances of getting the dynamic
specification right remain slim (Adolph, Butler, and Wilson Reference Adolph,
Butler and Wilson2005; Plümper, Troeger, and Manow Reference Plümper, Troeger
and Manow2005; Wilson and Butler Reference Wilson and Butler2007; DeBoef and
Keele Reference DeBoef and Keele2008). Our findings also suggest that a
difference in pooled-OLS and fixed-effects estimates cannot with certainty be
attributed to time constant unit heterogeneity.Footnote 5 It may equally be
caused by, inter alia, omitted time-varying variables, wrong assumptions about
the functional forms of the treatment effect, and misspecified lag structures.
The article studies the consequences of dynamic misspecification that occur in
static models or when applied researchers use simple econometric patches instead
of a correct dynamic specification. Our formal and simulation analyses support
previous arguments that fixed-effects estimates are biased if the model suffers
from excluded time-varying variables and if trends and dynamics are not
correctly modeled. We also demonstrate that the widely shared assumption, the
fixed-effects model is superior to our naïve benchmark, pooled-OLS, does not
necessarily hold in the presence of dynamic misspecifications. We provide
evidence that under identifiable and plausible conditions the fixed-effects
estimator may actually exacerbate the bias in comparison to a naïve estimator
even in presence of omitted time-invariant variables, because dropping the
between variation increases the influence of dynamic misspecifications on
parameter estimates.
We examine the overall logic of dynamic misspecifications based on three simple
examples: the existence of omitted time-invariant and time-varying variance
(experiment 1), trends in both an omitted variable and the variable of interest
(experiment 2), and a misspecified lag structure of the explanatory variable of
interest (experiment 3). Our results confirm that the fixed-effects estimator is
biased in the presence of omitted variables which either vary over time or exert
a time-lagged effect on the outcome—a result known from theoretical work (Lee
Reference Lee2012; Ahn, Lee, and Schmidt Reference Ahn, Lee and Schmidt2013). We
go one step further and demonstrate that using fixed effects in the presence of
time-varying and time-invariant omitted variables can under plausible
assumptions increase the bias relative to a naïve estimation with pooled-OLS or
the random-effects estimator. Our results also invalidate the common
interpretation of the Hausman test, namely that if the fixed and random-effects
(or pooled-OLS) estimates significantly differ, then researchers should use the
(consistent) fixed-effects model (Hausman Reference Hausman1978; Baltagi
Reference Baltagi2001, 65–70). This interpretation of the Hausman test assumes
the absence of any other model misspecification that influences fixed effects
and pooled-OLS estimates differently.Footnote 6 The results of our analyses
bring a common problem to attention: econometric solutions to a single
specification issue can impede the accuracy of estimates even though the
econometric patch solves the problem it has been invented for. For example, the
fixed-effects estimator has been developed to eliminate bias from ‘unobserved
heterogeneity’Footnote 7 due to constant unit-specific effects, but by doing so
it can amplify the bias resulting from un-modeled dynamics.Footnote 8 Our
findings stress the importance of developing model specifications for multiple
simultaneous model misspecifications. Biases generated by different model
misspecifications are often not additive, which implies that solving one problem
can exacerbate the bias emanating from another misspecification.Footnote 9
2 THE SOURCES OF AND POTENTIAL FOR DYNAMIC MISSPECIFICATION
Applied researchers often perceive serially correlated errors as noise rather
than information (DeBoef and Keele Reference DeBoef and Keele2008). Yet,
serially correlated errors clearly indicate a potentially severe model
misspecification, which can result from various sources (Neumayer and Plümper
Reference Neumayer and Plümper2017). Perhaps most obviously, serially correlated
errors are caused by incompletely or incorrectly modeled persistency in the
dependent variable, time-varying omitted variables or changes in the effect
strengths of time-invariant variables, or misspecified lagged effects of
explanatory variables. Conditionality makes modeling dynamics more complicated
(Franzese Reference Franzese2003a,Reference Franzeseb; Franzese and Kam
Reference Franzese and Kam2009). Few empirical analyses model all potential
conditioning factors of the variables of interest. If, however, treatment
effects are conditioned by unobserved time-varying factors—as for example the
effect of higher education on income is conditioned by structural change of the
economy and ruptures in economic policies—then treatment effects vary over time,
and the strength of these effects also changes over time as un-modeled
conditioning factors change. Finally, serially correlated errors may result from
misspecification that at first sight have little to do with dynamics, for
example from spatial dependence. Yet, spatial effects are certainly
misunderstood if they are perceived as time-invariant, ignoring spatial
dependence causes errors to be serially correlated (Franzese and Hays Reference
Franzese and Hays2007). Virtually all of these complications depend on an
arbitrary decision that no researcher can avoid: the periodization of continuous
time that is a necessary condition if researchers wishing to study ‘periods’. If
researchers choose relatively short periods, effects do no longer necessarily
occur in the same period as the treatment. If periods cover a long stretch of
time, the probability that estimates are biased by confounders rises quickly. In
the social sciences, the lengths of a period is rarely chosen to optimize the
analysis. Instead, social scientists often have to accept data that is collected
on a daily, monthly or—most often—annual basis.
At least in an optimal world these model misspecifications should be avoided:
dynamics should be directly modeled to obtain unbiased estimates. This proves to
be difficult. Since dynamic misspecifications are manifold and complex,
econometric tests for ’dynamics’ at best reveal serially correlated errors, but
they are usually unable to identify the underlying root causes of
autocorrelation. Often, these tests are also weak and do not reveal the true
dynamic structure of the data-generating process, which may lead to overfitting
of the data (Keele, Linn, and Webb Reference Keele, Linn and Webb2016). Thus,
empirical researchers are probably best advised to simplify their empirical
model and to treat problems such as serially correlated errors with
straightforward econometric patches such as lagged dependent variables, period
dummies, and simple homogeneous lag structures.
Yet, using misspecification patches should not mislead researchers into
believing that the dynamics of their model are correctly specified. Econometric
fixes are not correct per se because they are usually not modeling the true
dynamic process in the underlying data-generating process. For example, periods
do not exert a direct effect on the dependent variable but period dummies
capture variation over time which can help to “clean” residuals. Perhaps even
worse, more than one econometric specification allows eliminating serial
correlation, and there is no guarantee that different models lead to identical
or at least sufficiently similar results. Empirical researchers should also not
expect that so called ‘dynamic econometric models’, e.g. the Arellano–Bond (A–B)
estimator (Arellano and Bond 1991), solve various problems of dynamics. Dynamic
panel models only eliminate Nickell-bias. Period dummies control for common
trends, common shocks, and common breaks, but they do not perfectly account for
unit-specific, heterogeneous trends, shocks, and breaks. Including a lagged
dependent variable to the right hand side of the estimation equation without
including lags of the explanatory variables ( xx ) assumes that the dynamics of
all independent variables are identical. These assumptions are convenient, but
not always plausible.
Still, the vast majority of panel data analyses pushes serially correlated
errors into uninformative econometric patches: lagged dependent variables and
period fixed effects appear to be the most common solutions, but they are not
the only ones. More often than not analysts seem to “adopt restrictive dynamic
specifications on the basis of limited theoretical guidance and without
empirical evidence that restrictions are valid, potentially biasing inferences
and invalidating hypothesis tests” (DeBoef and Keele Reference DeBoef and
Keele2008, 184).Footnote 10 A review of recent political science publications
reveals that a large majority of panel data analyses rely on of the following
four strategies: do nothing and ignore the potential for dynamics (Humphreys and
Weinstein Reference Humphreys and Weinstein2006; Ross Reference Ross2008),
assume that all dynamics are captured by period fixed effects (Egorov, Guriev,
and Sonin Reference Egorov, Guriev and Sonin2009; Besley and Reynal-Querol
Reference Besley and Reynal-Querol2011; Menaldo Reference Menaldo2012 among many
others), try to capture dynamics by a lagged dependent variable (e.g. Acemoglu
et al. Reference Acemoglu, Johnson, James, Robinson and Yared2008; Guisinger and
Singer Reference Guisinger and Singer2010; Lupu and Pontusson Reference Lupu and
Pontusson2011; Kogan, Lavertu, and Peskowitz Reference Kogan, Lavertu and
Peskowitz2016), or finally follow Beck and Katz (Reference Beck and Katz1995)
and model dynamics by a combination of period fixed effects and a lagged
dependent variable (e.g. Lipsmeyer and Zhu Reference Lipsmeyer and Zhu2011;
Getmansky and Zeitzoff Reference Getmansky and Zeitzoff2014). Significantly
fewer authors rely on GLS estimators (Mukherjee, Smith, and Li Reference
Mukherjee, Smith and Li2009; Lupu and Pontusson Reference Lupu and
Pontusson2011), distributed lag models (e.g. Gerber et al. Reference Gerber,
Gimpel, Green and Shaw2011) or error correction models (Lebo, McGlynn, and Koger
Reference Lebo, McGlynn and Koger2007; Kayser Reference Kayser2009; Soroka,
Stecula, and Wlezien Reference Soroka, Stecula and Wlezien2015).Footnote 11
Overall, the vast majority of panel data analyses in political science assumes
rather simple dynamics.Footnote 12 This finding is consistent with DeBoef and
Keele (Reference DeBoef and Keele2008, 185), who also conclude that the vast
majority of authors do not test for the underlying dynamic structure. Thus,
social scientists often model the dynamic aspects with very little theoretical
guidance (Keele and Kelly Reference Keele and Kelly2006; DeBoef and Keele
Reference DeBoef and Keele2008),Footnote 13 use ad hoc econometric solutions,
which make rather rigid assumptions, do not try to model the true
data-generating process, and do not report results of minimal tests for serial
correlation.
One strategy that may reduce the size of the problem is to use less constrained
econometric solutions. Distributed lag models, models with a unit-specific
lagged dependent variable, panel co-integration models, models with
heterogeneous lag structure (Plümper, Troeger, and Manow Reference Plümper,
Troeger and Manow2005), more attention to periodization (Franzese Reference
Franzese2003a), better specified spatial models (Franzese and Hays Reference
Franzese and Hays2007; Neumayer and Plümper Reference Neumayer and Plümper2016)
may all reduce the size of the problem. However, as the number of possible
dynamic specifications increases, a higher order problem of model selection
arises: since all these different models likely generate different estimates and
often demand different inferences, the question becomes how empirical
researchers select their preferred model. To eliminate or at least reduce the
arbitrariness of model selection, DeBoef and Keele (Reference DeBoef and
Keele2008, 187) suggest a testing-down approach, starting with a full
autoregressive distributive lag model and stepwise removing parameters according
to pre-determined criteria, often the significance of parameters. This procedure
will result in a dynamic specification that maximizes the variance absorbed by
the minimum number of parameters. As with all testing-down approaches, this
approach suffers from the arbitrariness in the choice of a start model because
we do not have an infinite number of degrees of freedom. DeBoef and Keele
recommend starting with a general autoregressive distributed lag (ADL) model.
They argue that this model has ‘no constraints’. Yet, the model still assumes a
homogeneous lag structure and it will run quickly out of degrees of freedom if
the number of controls is large because a finite number of distributed lag
parameters has to be estimated for each regressor. Accordingly, these models
only work if the number of periods is much larger than the number of variables—a
criterion that is not necessarily met in panel data analyses. Since the
specification includes a lagged dependent variable, the estimator is
inconsistent when unit fixed effects are included, though the bias declines if
the number of periods increases (Nickell Reference Nickell1981; Kiviet Reference
Kiviet1995). Gerber et al. (Reference Gerber, Gimpel, Green and Shaw2011) thus
prefer an alternative strategy. Rather than relying on a single ‘best’ dynamic
specification, they report the result of various different dynamic specification
and they demonstrate that their results are robust “for varying lag lengths and
polynomial orders” (Gerber et al. Reference Gerber, Gimpel, Green and Shaw2011,
143). Relying on robustness tests has at least two advantages: first, it largely
reduces the necessity to make arbitrary dynamic modeling assumptions, and,
second, it helps identifying possible relevant model uncertainties (Neumayer and
Plümper Reference Neumayer and Plümper2017).
For our purposes and in the remainder of this article, the problem is not so
much which techniques minimizes the potential for dynamic misspecification.
Instead, we assume that dynamic misspecifications exist and analyze the
performance of the fixed-effects model in the presence of various dynamic
misspecifications—some of which could be dealt with easily, others are more
difficult though not impossible to eliminate if only researchers knew the true
data-generating process. But of course the whole point of estimation is that
researchers do not know the true data-generating process and that theory,
econometric tests, and testing-down procedures cannot identify the optimal model
beyond reasonable doubt. Having said this, we do not claim that social
scientists inevitably misspecify dynamics, but we emphasize that in the presence
of dynamic misspecifications, the fixed-effects model has problematic
properties. Needless to say that modeling dynamics correctly is always
preferable.
3 THE BIAS OF THE FIXED-EFFECTS ESTIMATOR WITH DYNAMIC MISSPECIFICATION
This section analyzes how dynamic misspecifications cause fixed-effects
estimates to be biased and we demonstrate that the bias of the fixed-effects
estimator can exceed the bias of the naïve pooled-OLS estimator under plausible
assumptions. We are not the first to do so. Lee (Reference Lee2012) analytically
demonstrates that the fixed-effects estimator is biased when the lag order is
not correctly chosen and stresses that “existing bias corrections would not work
properly because the correction formulas assume correct model specification. In
fact, attempts to adjust for the bias using formulas that correct for AR(1)
dynamics would be wrong and may even exacerbate the bias when the true lag order
is larger than one” (Lee Reference Lee2012, 57).
Misspecified lag structures are clearly not the only dynamic misspecification
that biases fixed-effects estimates. Rather, fixed-effects estimates are likely
to be biased in the presence of any dynamic misspecification or omitted
time-varying variables.Footnote 14 Ahn, Lee, and Schmidt (Reference Ahn, Lee and
Schmidt2013) argue that the fixed-effects estimator is biased when omitted
variables vary over time and develop a generalized method of moments procedure
that accounts for multiple factorial time-varying fixed effects. This estimator,
however, requires the existence of instruments which are correlated with the
dynamic fixed effects but not with the errors—an assumption that is unlikely to
be satisfied and that cannot be tested properly since errors remain unobserved.
Finally, Park (Reference Park2012) at least implicitly confirms the existence of
bias in fixed-effects models with structural breaks and develops a Bayesian
estimator that seeks to identify these structural breaks. As one would expect, a
model that corrects for ‘turning points’ fits the data better than the classical
fixed-effects estimator. We build on these contributions and prove that the bias
from dynamic misspecifications can be larger for the fixed-effects estimator
than for pooled-OLS. As we have already mentioned, we make this comparison not
so much because we intend to rehabilitate the pooled-OLS estimator. Rather, we
use this comparison to demonstrate how poorly the fixed-effects estimator
performs in the presence of dynamic misspecifications.
3.1 BIAS OF THE FIXED-EFFECTS ESTIMATOR INDUCED BY CORRELATED WITHIN VARIANCE
Fixed-effects estimation accounts for potential bias from unobserved
time-invariant variables by eliminating all between variation from the
estimation. Obviously, the effect of variance that is dropped from the
estimation cannot be biased by correlated confounders. And the correlation
between the remaining within variation and omitted time-invariant
cross-sectional variation is zero. Therefore, if the effect of omitted variables
is really exclusively time-invariant, the estimates which rely on an analysis of
the remaining time-varying variance does not suffer from omitted variable bias.
This, of course, immediately changes when the aggregate effect of omitted
variables is not strictly time-invariant.
In this section, we derive the causes of the bias of the dynamic fixed-effects
estimator using a time-varying omitted variable as an example. We demonstrate
that the bias of the fixed-effects estimate of β𝛽 exceeds the bias of the
pooled-OLS estimate of β𝛽 when the correlation between xx and omitted within
variation is larger than the correlation between xx and omitted between
variation.
Assume that
(1)
yit=α+βxit+ui+εityit=𝛼+𝛽xit+ui+𝜀it
is the true data-generating process with xitxit a time-varying observed
variable, uiui a vector of time-invariant unobserved variables, and εit𝜀it an
i.i.d. error component. Note that the data-generating process is static.
Estimating this model by a ‘naïve’ OLS estimator leads to bias if
var(x¯i,ui)≠0var(x¯i,ui)≠0 , or var(xit,εit)≠0var(xit,𝜀it)≠0 .
The fixed-effects estimator eliminates the between variation from Equation (1)
so that
(2)
yit−y¯i=β(xit−x¯i)+ui−u¯i+εit−ε¯iyit−y¯i=𝛽(xit−x¯i)+ui−u¯i+𝜀it−𝜀¯i
which is equivalent to
(3)
yit−y¯i=β(xit−x¯i)+εit−ε¯i,yit−y¯i=𝛽(xit−x¯i)+𝜀it−𝜀¯i,
because ui−u¯i=0ui−u¯i=0 .
Assume now the following data-generating process
(4)
yit=α1xit+α2wit+ui+εit;with α2=1yit=𝛼1xit+𝛼2wit+ui+𝜀it;with 𝛼2=1
where xitxit and witwit are time-varying right-hand-side variables and uiui is a
unit-specific effect.
The omitted variable witwit is correlated with the included right-hand-side
variable xitxit . γ1𝛾1 and γ2𝛾2 indicate the strength of the correlation
between witwit and the within variance of xitxit and witwit and the between
variation of xitxit , respectively:
(5)
wit=γ1x¨it+γ2x¯i+ωitwit=𝛾1x¨it+𝛾2x¯i+𝜔it
with x¯i=(1/T)∑Tt=1xit,x¨it=(xit−x¯i)x¯i=(1/T)∑t=1Txit,x¨it=(xit−x¯i) .
Finally, the unit-specific effect uiui covaries with the between variance of
xitxit to a degree of delta.
(6)
ui=δ1x¯i+νi.ui=𝛿1x¯i+𝜈i.
We omit witwit from the estimation and can easily derive the biases for the
fixed effects and the pooled-OLS estimators ( α^1,FE𝛼^1,FE and α^1,OLS𝛼^1,OLS
) under the assumptions in (4)–(6). We also can demonstrate that under certain
conditions the bias of fixed-effects estimates exceeds that of pooled-OLS
estimates. Needless to say that neither of these two estimators is unbiased in
case of time-varying omitted variables.Footnote 15
Conditional on all of the xitxit , Equation (7) derives the bias for the
pooled-OLS estimator:
(7)
Bias(α^1,OLS)=γ1∑i=1N∑t=1Tx¨2it+Tγ2∑i=1Nx¯2i∑i=1N∑t=1Tx¨2it+T∑i=1Nx¯2i+δ1T∑i=1Nx¯2i∑i=1N∑t=1Tx¨2it+T∑i=1Nx¯2i.Bias(𝛼^1,OLS)=𝛾1∑i=1N∑t=1Tx¨it2+T𝛾2∑i=1Nx¯i2∑i=1N∑t=1Tx¨it2+T∑i=1Nx¯i2+𝛿1T∑i=1Nx¯i2∑i=1N∑t=1Tx¨it2+T∑i=1Nx¯i2.
Equation (7) indicates that the OLS bias depends both on the correlation between
xitxit and witwit as well as the correlation between uiui and xitxit .
As usual the bias for the FE estimator is given by:
(8)
Bias(α^1,FE)=γ1.Bias(𝛼^1,FE)=𝛾1.
The bias of the fixed-effects estimator depends on the correlation between
xitxit and witwit , but not on the covariance between the unit-specific effects
uiui and xitxit , because the within-transformation on which FE estimation
relies, effectively eliminates all, endogenous and exogenous, between variation
from the estimation.
If we assume that δ1=0𝛿1=0 (no correlation between uiandx¯iuiandx¯i ) and
γ2=0𝛾2=0 (no correlation between witandx¯iwitandx¯i ), then,
(9)
Bias(α^1,OLS)=γ1∑i=1N∑t=1Tx¨2it∑i=1N∑t=1Tx¨2it+T∑i=1Nx¯2i<γ1=Bias(α^1,FE).Bias(𝛼^1,OLS)=𝛾1∑i=1N∑t=1Tx¨it2∑i=1N∑t=1Tx¨it2+T∑i=1Nx¯i2<𝛾1=Bias(𝛼^1,FE).
In this case, for any given T<∞T<∞ , the bias of the FE estimator that results
from the omission of witwit is larger than that of OLS. This is so because the
fraction term of the OLS bias in Equation (9) is always smaller than 1.
This case might seem rare in real data but can emerge when neither xitxit nor
witwit have a specific dynamic structure (autocorrelation or trends) but only
the variation over time and not across units of these two variables is related.
An exogenous shock could have this property. Alternatively, witwit has no
between variation and represents an omitted common trend. More often, however,
applied researchers specify empirical models that suffer from both, omitted
between variation correlated with the regressors and omitted within variation
correlated with the regressors. In these cases, one cannot say whether the fixed
effects or the pooled-OLS estimator gives less biased estimates. One can ex ante
know that both estimators give biased results, but which one is more reliable
(or less unreliable) depends on the relative strengths of the correlations with
the omitted variance. Unfortunately, these correlations cannot be observed.
Researchers may often know that a relevant variable has been omitted, but one
cannot know with certainty that no relevant variable has been omitted. Still,
one can evaluate whether omitted variables are potentially more problematic for
the included within or between variation by estimating how much of the within
and between variance of the dependent variable remains unexplained.
Now consider a situation where xitxit (and witwit ) follows a deterministic
trend so that its within variance grows with increasing number of time periods,
and approaches infinity asT→∞asT→∞ . In this case, even if δ1≠0𝛿1≠0 (non-zero
correlation between uiandx¯iuiandx¯i ) the second term of the OLS bias Equation
(7) approaches zero because the within variation ( x¨2itx¨it2 ) grows but only
appears in the denominator while the between variance ( x¯2ix¯i2 ) does not
change. The bias that is caused by the correlation between witwit and xitxit
increases with TT because of the trend and will outweigh the bias induced by
omitted time-invariant variables if TT grows large enough.
3.2 BIAS FROM DYNAMIC MISSPECIFICATION
Correlated within variation and common trends of included and excluded
explanatory variables are obvious sources of omitted variable bias occurring in
fixed-effects estimates. Yet, there are many examples of dynamic
misspecifications that can cause bias. Assume a data-generating process
representing the simplest form of dynamic misspecification, an explanatory
variable that does not exert a contemporaneous effect on the dependent variable
but a one period lagged effect:
(10)
yit=βxit−1+ui+εit.yit=𝛽xit−1+ui+𝜀it.
If we estimate Equation (10) ignoring the lagged effect of xitxit , the
probability limit (plim) of the OLS estimator of β𝛽 in the regression
yit=βxit+εityit=𝛽xit+𝜀it is given by:
(11)
Cov(yit,xit)Var(xit)=βCov(xit−1,xit)Var(xit)+Cov(ui,xit)Var(xit).Cov(yit,xit)Var(xit)=𝛽Cov(xit−1,xit)Var(xit)+Cov(ui,xit)Var(xit).
The second term of Equation (11) is similar to the bias of estimating a model
without fixed effects while the true DGP has correlated unit effects: the
estimated β𝛽 wrongly captures the unit-specific effects (unless x¯i=0x¯i=0 ).
The probability limit of the fixed-effects estimator equals:
(12)
Cov(yit,x¨it)Var(xit)=βCov(xit−1,x¨it)Var(xit)+Cov(ui,x¨it)Var(xit).Cov(yit,x¨it)Var(xit)=𝛽Cov(xit−1,x¨it)Var(xit)+Cov(ui,x¨it)Var(xit).
The second term now vanishes since x¨itx¨it has no unit-specific mean. We can
rewrite Equation (12) so that
(13)
Cov(yit,x¨it)Var(xit)=βCov(xit−1,xit)Var(xit)−βCov(xit−1,x¯i)Var(xit)Cov(yit,x¨it)Var(xit)=𝛽Cov(xit−1,xit)Var(xit)−𝛽Cov(xit−1,x¯i)Var(xit)
(14)
⇔Cov(yit,x¨it)Var(xit)=βCov(xit−1,xit)Var(xit)−βVar(x¯i)Var(xit).⇔Cov(yit,x¨it)Var(xit)=𝛽Cov(xit−1,xit)Var(xit)−𝛽Var(x¯i)Var(xit).
The second term of Equation (14) equals β𝛽 multiplied by the between variance
of xitxit divided by its total variance. The result will fall between 0 and 1.
It follows that the probability limit of the within estimator (FE) is smaller
than β𝛽 . The estimate will thus be downward biased and this bias increases as
the share of ignored between variation in xitxit increases.
The total bias of the OLS estimator depends on the autocorrelation of xitxit and
the bias induced by the omission of the unit-specific effects. If the majority
of autocorrelations in real world data generation processes is positive (which
seems to be the case), the bias of a fixed-effects estimator exceeds the bias of
pooled-OLS. It is of course possible to estimate whether xit−1andxitxit−1andxit
are positively correlated and how strong this correlation is. However, it is
much more complicated to identify the correct lag structure of explanatory
variables (Adolph, Butler, and Wilson Reference Adolph, Butler and Wilson2005;
Plümper, Troeger, and Manow Reference Plümper, Troeger and Manow2005). Time
series tests such as information criteria (BIC, AIC etc.) have low power in
complex models and usually predict diverging lag lengths depending on the number
of lags and right-hand-side variables included. The problem of misspecified lag
length is exacerbated if the lag length is not uniform but varies across units
which can occur frequently in political science data, for example because
institutional settings will usually influence responsiveness of actors (Plümper,
Troeger, and Manow Reference Plümper, Troeger and Manow2005). We analyze the
effect of unit-specific lag length in the Monte Carlo experiments below.
3.3 DISCUSSION
We have demonstrated that biases from two different sources of model
misspecification are not simply additive. Rather, the solution to one problem,
time-invariant omitted variables, can easily make another problem, say omitted
time-varying variables, worse. In the following section we use Monte Carlo
analyses to compare the bias of the fixed-effects estimator to the bias of the
estimator that econometricians call naïve, pooled-OLS.Footnote 16 We do so to
identify some of the conditions under which the fixed-effects estimator has poor
properties. As we have mentioned before, we use pooled-OLS to have a benchmark
for ‘poor properties’—and not to recommend the choice of the pooled-OLS
estimator in applied research.
4 DESIGN OF THE MONTE CARLO EXPERIMENTSFOOTNOTE 17
Bias in fixed-effects estimation can result, inter alia, from omitted
time-varying variables, from omitted trends, a misspecified lag structure, and
other—more complex—dynamic misspecifications. Since social scientists often rely
on standard dynamic specifications rather than on explicitly modeling the
dynamics, bias may be reduced, but is unlikely to disappear. As we have shown in
the previous section, the existence of any form of unaccounted within variation
correlated with the regressors biases fixed-effects estimates. The Monte Carlo
analyses in this section aim at exploring the relevance of the problem. To
benchmark the bias of the fixed-effects estimator, we use the pooled-OLS
estimator which is known to have poor properties in the presence of omitted
time-invariant variables and dynamic misspecifications. Naïvely, one could
expect that, since pooled-OLS suffers from (at least) two problems while the
fixed-effects estimator solves the problem of omitted time-invariant variables,
the bias of the fixed-effects estimator is always strictly smaller than the bias
of pooled-OLS. However, this perspective ignores the fact that the fixed-effects
estimator solely uses the within variation and is therefore more vulnerable to
dynamic misspecification than pooled-OLS that uses both, the within- and the
between variation. As we demonstrate analytically, it is thus possible that
fixed-effects estimates are more biased than the pooled-OLS estimates under
identifiable conditions. To study the properties of the fixed-effects estimator
with potential dynamic misspecification and to reveal the conditions under which
the use of fixed effects produces larger bias than the naïve pooled-OLS
estimator, we employ a set of Monte Carlo experiments.
Our data-generating process follows a straightforward set-up:
(15)
yit=x1it+(x2it)+ui+εityit=xit1+(xit2)+ui+𝜀it
with x1itxit1 , x2itxit2 , εit𝜀it , and uiui being drawn from a standard normal
probability density function.
We use three rather straightforward types of model misspecifications as
examples: an omitted time-varying variable, an omitted time trend when the
variable of interest x1itxit1 is trended, and the simple dynamic
misspecification analyzed formally in the previous section—a one period lagged
effect of x1itxit1 . We distinguish three levels of correlation between our
variable of interest x1itxit1 and an omitted strictly time-invariant, constant
effect variable uiui . We set this correlation between x1itxit1 and uiui to 0.0
(absent), 0.2 (weak), and 0.5 (substantive). Higher correlation between x1itxit1
and uiui implies higher bias of the pooled-OLS estimator, while the correlation
between x1itxit1 and uiui does not bias the fixed-effects estimator. The higher
the correlation between x1itxit1 and uiui , the larger the bias advantage of the
fixed-effects model before we consider a dynamic misspecification. Obviously, in
the absence of dynamic misspecification the fixed-effects estimates are
unbiased. Throughout all specifications we assume that between and within
effects are equal. We acknowledge that this is a strong assumption and that
pooled-OLS gives an average estimate of the two effects while the fixed-effects
estimator provides a clean estimate for the within effect only. For a discussion
of dealing with different within and between effects see Bell and Jones
(Reference Bell and Jones2015). We refrain from adding a discussion of different
effects across units and over time since it would distract from the focus on
bias stemming from dynamic misspecification.
We are of course aware that social scientists could be able to correctly model
these simple dynamic misspecifications. But this argument misses the point: we
do not seek to identify dynamic misspecifications which are so difficult to
model that social scientists probably fail to fully eliminate them. Instead, we
are analyzing the consequences of dynamic misspecifications. The advantage of
simple dynamic misspecifications, thus, is that it is easy to understand how
they bias the fixed-effects estimator. Only in a second step will we generate
complex data-generating processes for which simple solutions are not available.
None of the data-generating processes we study here are likely to be as complex
as true data-generating processes. Given that we include simple dynamics, we do
not just use a simple fixed-effects specification, but rather compare
fixed-effects estimation with dynamic specifications that applied researchers
are likely to use as econometric solutions for potential dynamic
misspecifications:Footnote 18 a lagged dependent variable (or Arellano–Bond
dynamic panel modelFootnote 19 ), the Prais–Winsten transformation, or period
fixed effects. This also allows us to demonstrate that these simple fixes, which
are widely used in panel and pooled analyses, do not sufficiently eliminate
simple dynamic misspecifications. In addition to these simple but commonly
employed dynamic fixes, we use more general dynamic specifications as offered by
ADL models and show that capturing the most salient dynamic elements of a DGP
can reduce the bias considerably. This is consistent with Pickup (Reference
Pickup2017).
For our first two experiments, x2itxit2 is the omitted part of the
data-generating process. We first directly manipulate the correlation between
the within variation of x1itxit1 and x2itxit2 and the unit heterogeneity, e.g.
the covariance of the between variance of x1itxit1 and the unobserved
unit-specific effects uiui with
corr(x¨1it,x¨2it)={0.2,0.5,0.8}corr(x¨it1,x¨it2)={0.2,0.5,0.8} .
The second set of experiments aims at demonstrating the logic of our argument
without ex ante assuming that x1itxit1 and x2itxit2 are correlated. We generate
a dynamic misspecification by merely trending both variables so that the
correlation results from the trends only. We discuss two different variants of
this second Monte Carlo experiment.Footnote 20 The first variant assumes common
trends across all units: Both included and excluded right-hand-side (RHS)
variables are continuous with a common trend of 0.1 increase per time period:
(16)
x1,2it=N∼(0,1)+0.1∗t,t=1,…,T.xit1,2=N∼(0,1)+0.1∗t,t=1,…,T.
The second variant relaxes this assumption and allows for unit-specific
trends,Footnote 21 which merely means that trends are conditioned by other
factors—a plausible assumption for social scientists, since trends are unlikely
to be homogeneous across units. Specifically, we randomly draw a third of the
units that receives a positive trend of 0.1 per time period (see Equation (16)),
a third of the units remains untrended ( x1,2it=N∼(0,1)xit1,2=N∼(0,1) ), and the
last third of units has a negative trend of 0.1 per time period
x1,2it=N∼(0,1)−0.1∗t,t=1,…,Txit1,2=N∼(0,1)−0.1∗t,t=1,…,T ).Footnote 22
The third experiment is based upon a slightly different DGP to account for a
misspecified lag structure of x1itxit1 :
(17)
yit=x1it−1+ui+εit.yit=xit−11+ui+𝜀it.
We compare the bias generated by a static OLS estimator (
yit=x1it+εityit=xit1+𝜀it ) to that of a static FE estimator (
y¨it=x¨1it+ε¨ity¨it=x¨it1+𝜀¨it ) where the lagged effect of x1itxit1 is not
taken into account.
Finally, we also allow the lag length of x1itxit1 to vary across units in the
following way: for one randomly drawn third of the units x1itxit1 exerts a one
period lagged effect on yityit as in Equation (17), for the second randomly
drawn third of units we observe a two period lagged effect
yit=x1it−2+ui+εityit=xit−21+ui+𝜀it and for the last third we model a three
period lagged effect yit=x1it−3+ui+εityit=xit−31+ui+𝜀it .
We vary the number of periods [ T={10,30,50}T={10,30,50} ] but we hold the
number of units constant at 20 throughout all experiments. Note that increasing
the number of units increases the between variation and favors pooled-OLS over
fixed effects (Plümper and Troeger Reference Plümper and Troeger2007, Reference
Plümper and Troeger2011). In each permutation of the experiments we estimate 500
models with independently drawn errors.
Since econometricians have developed different solutions for models with
potential dynamic misspecifications, we incorporate these variants of the fixed
effects and the pooled-OLS estimators into the simulation. The most commonly
used ‘solutions’ to dynamic misspecification are the inclusion of the lagged
dependent variable (LDV: yit=α+β1yit−1+β2x1it+εityit=𝛼+𝛽1yit−1+𝛽2xit1+𝜀it
),Footnote 23 and period fixed effects ( yit=αt+β2x1it+εityit=𝛼t+𝛽2xit1+𝜀it )
or a combination of the two (
yit=αt+β1yit−1+β2x1it+εityit=𝛼t+𝛽1yit−1+𝛽2xit1+𝜀it ). Less often researchers
employ a Prais–Winsten transformation (PW:
(yit−ρyit−1)=α+β2(x1it−ρx1it−1)+(εit−ρεit−1)(yit−𝜌yit−1)=𝛼+𝛽2(xit1−𝜌xit−11)+(𝜀it−𝜌𝜀it−1)
),Footnote 24 or an ADL ((1,1):
yit=α+β1yit−1+β2x1it+β3x1it−1+εityit=𝛼+𝛽1yit−1+𝛽2xit1+𝛽3xit−11+𝜀it ) model.
Though it seems to increasingly be the case that scholars estimate fixed-effects
models without justification and thus as default, econometric textbooks suggest
a variant of the Hausman specification test (Hausman Reference Hausman1978) to
decide whether to estimate a fixed effects or a random-effects/ OLS
specification. The Hausman test (and its variants) have been shown to be
consistent (for a short overview see Baltagi Reference Baltagi2001, 65–70),
therefore, if fixed-effects estimates are significantly different from random
effects or pooled-OLS estimates, the latter are biased because of unit
heterogeneity.Footnote 25 However, the asymptotic properties of the Hausman test
do not necessarily translate into favorable finite sample characteristics
especially when other misspecifications do exist and are not accounted for. We
also present Monte Carlo results for the performance of this test. This is
related to our main research interest, because we intend to demonstrate that
pooled-OLS may be less biased than the fixed-effects model in situations in
which the Hausman test favors a fixed-effects specification. These instances may
occur frequently and under plausible conditions.
5 RESULTS
Applied researchers should select estimators according to their reliability for
the sample at hand. The root mean squared error has been suggested as the
appropriate criterion for selecting estimators in finite samples. The root mean
squared error provides information on the average deviation of an estimator from
the true relationship. This average deviation results from bias and sampling
variation of an estimator. We show the bias for our MC experiments because
whenever the bias of the fixed-effects estimates exceeds the bias of the
benchmark, the pooled-OLS estimator, the root mean squared error is also larger.
Since OLS is using both within and between variation for estimation it is the
more efficient estimator as compared to Fixed Effects.
We run five sets of experiments that examine different dynamic
misspecifications: (i) omitted time-varying variable, (ii) omitted common trend,
(iii) unit-specific trend, (iv) misspecified common lag structure, and (v)
misspecified unit-specific lag structure. For each of the misspecifications we
estimate six different fixed effects and pooled-OLS models with different
dynamic specification: no dynamics, lagged dependent variable (LDV, or
Arellano–Bond (A–B) model), Prais–Winsten GLS transformation, period fixed
effects, a combination of LDV/A–B and period fixed effects, and an ADL (1,1)
model. Finally we vary the correlations between the unit-specific effects uiui
and the interesting RHS variable x1itxit1 (as described above), as well as the
number of periods.
Table 1 summarizes the findings of all conducted experiments. We show the
average, minimum and maximum bias generated by pooled-OLS and the fixed-effects
model for each dynamic specification.
Table 1. Bias over all Experiments.
Table 1 gives a first impression of the general performance of pooled-OLS and
fixed-effects models with different econometric patches when dynamic
misspecifications are present in the DGP but not necessarily properly accounted
for in the specification of the estimation equation. Overall, the average bias
of the coefficient for x1itxit1 (the RHS variable of interest) produced by
pooled-OLS is up to 45 percent smaller than that generated by the fixed-effects
estimator. In addition, the maximum bias of OLS is usually considerably smaller
than the maximum bias of the fixed-effects estimates (except when a
Prais–Winsten GLS transformation is applied). An ADL(1,1) model estimates
coefficients for both x1itxit1 and the one period lagged x1it−1xit−11 . The ADL
model produces on average less biased estimates for x1it−1xit−11 when unit fixed
effects are included. However, the computed average bias for estimates of
x1itxit1 and x1it−1xit−11 in the ADL(1,1) model is somewhat misleading because
in experiments 1 and 2 x1itxit1 should be included in the estimation but
x1it−1xit−11 is not part of the DGP, while in experiment 3 only x1it−1xit−11 has
an effect on the outcome. Three of the dynamic specifications we test (LDV, LDV
++ period FE, ADL) also estimate coefficients for yit−1yit−1 . This coefficient
should be zero because yit−1yit−1 is never part of the DGP. Specifications that
include unit fixed effects on average produce coefficients for the LDV that are
closer to zero. In a pooled-OLS specification the LDV on average seems to pick
up potential unit-specific effects that remain un-modeled.Footnote 26 If
researchers are interested in persistency of the dependent variable or long term
effects and unit effects are indeed present, a fixed-effects specification
produces less biased estimates of the LDV coefficient. This often comes at the
expense of a more biased estimate for the interesting explanatory variables when
dynamic misspecifications are present. To unpack the relative performance of
both estimators in the presence of different dynamic misspecifications we
present disaggregated results for each misspecification and different
econometric controls for dynamics.
5.1 EXPERIMENT 1: OMITTED TIME-VARYING VARIABLE
We start with examining the effect of omitted time-varying variables for
different levels of correlated unit-specific heterogeneity. The results confirm
the theoretical results in Section 3. Table 2 depicts the bias of OLS (solid
line) and FE (dashed line) with an assumed within correlation between included
and omitted time-varying variables of 0.5. We include the results for eighteen
combinations for the level of correlation of x1itxit1 and uiui and a dynamic
specification. Each single figure displays the bias for the OLS estimates and
the bias for the fixed-effects estimates (right axis) plus the probability that
the Hausman test finds a significant difference between the OLS and the FE
estimates (at the 95 percent level—gray shaded area, left axis). The larger the
gray shaded area, the higher the probability that the Hausman test recommends
the FE model. We show results for each of the six specifications that political
scientists frequently use to control for dynamics: no control for dynamics,
lagged dependent variable (with Arellano–Bond estimator—dotted lineFootnote 27
), Prais–Winsten transformation, period fixed effects, the combination of the
LDV and period fixed effects, and an ADL specification. For the ADL(1,1) model
(last specification in each table) we display bias for estimates of x1itxit1
(black lines) and x1it−1xit−11 (gray lines). The columns depict these results
for different levels of correlation between the unit-specific effects uiui and
the included treatment x1itxit1 .
Table 2. Omitted Within Variance corr(x¨1it,x¨2it)=0.5corr(x¨it1,x¨it2)=0.5 :
Bias for Estimate of x1itxit1 and x1it−1xit−11 .
Note: Right Axis—Absolute Bias: —— OLS, - - - - - FE, ⋯⋯⋯⋯ A–B (ADL: gray lines
== bias of coefficient for x1it−1xit−11 ); Left Axis—Probability of rejecting
the H0 on the 5% level and thus suggesting FE: gray shaded area == Hausman Test.
Table 2 illustrates that the bias of the fixed-effects model increases as the
correlation between the variable of interest and an omitted time-varying
variable increases (see tables A1 and A2 in appendix for comparison). The
fixed-effects estimator is not immune against different sources of unobserved or
omitted heterogeneity, it merely shelters estimates from omitted time-invariant
variables with constant effects (which is referred to as ‘unobserved
heterogeneity’ in most econometric textbooks). The bias of the fixed-effects
estimates remains unaffected by changes in the correlation between the variable
of interest and an omitted time-invariant variable.
The omission of a time-varying variable that is correlated with included
right-hand-side variables may lead to serially correlated errors and it will
induce bias. As we have explained in Section 2, social scientists use various
econometric solutions to control for the serial correlation of errors
potentially resulting from omitted time-varying variables. We find that these
solutions have virtually no effect on the bias of the fixed-effects estimate in
the presence of omitted time-varying variables. Yet, omitted time-varying
variables are a common problem in the social sciences—arguably more common than
the omission of variables with time-invariant effects that vary across units.
A comparison between the properties of the fixed-effects model and the benchmark
pooled-OLS estimator reveals that the fixed-effects model is more (less) biased
if the correlation of the variable of interest with the omitted within variation
is larger (smaller) than the correlation with the omitted between variation. Of
course, if no omitted time-invariant variable exists but the model is
dynamically misspecified, pooled-OLS is strictly less biased than the
fixed-effects estimator. This confirms the results from Section 3. The results
for the ADL(1,1) model show the same bias differential between OLS and FE
estimates for x1itxit1 (black lines), though the difference is smaller. However,
a fixed-effects specification seems to be able to deal much better with elements
that are not included in the DGP since it produces a much smaller bias for the
unnecessarily added x1it−1xit−11 (gray lines).
Table 2 also reveals the low power of the Hausman test in the presence of
dynamic misspecification. It gives erratic results and in the worst case with no
omitted time-invariant variables but a correlated omitted time-varying variable,
the Hausman test always suggests the use of the fixed-effects model—even if no
omitted between variation exists. We also find that the Hausman test is
sensitive to the choice of dynamic specification. If applied researchers include
a lagged dependent variable, the Hausman test is biased toward the fixed-effects
model—a finding that confirms previous research (Arellano Reference
Arellano1993; Ahn and Low Reference Ahn and Low1996; Godfrey Reference
Godfrey1998; Baltagi Reference Baltagi2001; Hoechle Reference Hoechle2007,
66–69). In other words, the ‘consistency’ of the Hausman test is conditional on
a perfectly specified model that suffers solely from omitted between variation
with constant unit effects.
Finally, in our MC analyses all results are largely independent of the number of
periods, because we hold the within correlation constant. If, in reality, adding
periods leads to a change in the correlation, the bias will also change. As
adding time periods increases the probability of correlated time-varying omitted
variables, the bias will increase over-proportionally for fixed-effects
estimates.
5.2 EXPERIMENT 2. CORRELATED COMMON AND UNIT-SPECIFIC TRENDS
In the second experiment, we study the bias of the fixed-effects model and the
pooled-OLS estimator when both the variable of interest x1itxit1 and an omitted
time-varying variable x2itxit2 are trended. Two trended variables tend to be
correlated even if they are independent of each other. Table 3a displays the
results for an excluded trended variable, while Table 3b provides the results
for experiments in which the trend is assumed to be unit-specific.
This experiment confirms that the static fixed-effects model is biased, but this
bias—expectedly—disappears when scholars include period fixed effects in the
presence of a true common trend. A similar result can be achieved by the
inclusion of splines, but period fixed effects follow the functional form of the
omitted trended variable more closely. Unfortunately, period fixed effects also
capture the trend of other trended variables. Hence, if scholars aim at
analyzing dynamic processes, period fixed effects only leave unit-specific
deviations from the common trend for variables of interest, since period fixed
effects account for all common trends. This does not mean that we suggest
leaving out period fixed effects in general, we advocate a less ad hoc approach
to modeling of the salient dynamic features of the data-generating process and a
far more cautious interpretation of the estimation results.
With omitted trended variables and no period FE, pooled-OLS tends to outperform
the fixed-effects model unless the number of periods remains small and the
correlation between a time-invariant omitted variable uiui and the variable of
interest x1itxit1 is high. As we have demonstrated in Section 3, trends increase
the within variation of included and omitted RHS variables when TT grows larger.
As a consequence, the bias resulting from omitted trends increases in TT , which
affects fixed-effects models more strongly than pooled-OLS models because FE
solely relies on within variation for estimation. This observation holds true
for the ADL (1,1) estimation of x1itxit1 . However, as in experiment 1,
including unit fixed effects allows estimating zero effects of unnecessary
components ( x1it−1xit−11 ) more precisely though not without bias.
Table 3a. Omitted common trend: bias for estimate of x1itxit1 and x1it−1xit−11 .
Note: Right Axis—Absolute Bias: —— OLS, - - - - - FE, ⋯⋯⋯⋯ A–B (ADL: gray
lines == bias of coefficient for x1it−1xit−11 ); Left Axis—Probability of
rejecting the H0 on the 5% level and thus suggesting FE: gray shaded area ==
Hausman Test.
Table 3b. Omitted unit-specific trends: bias for Estimate of x1itxit1 and
x1it−1xit−11 .
Note: Right axis—absolute bias: —— OLS, - - - - - FE, ⋯⋯⋯⋯ A–B (ADL: gray lines
== bias of coefficient for x1it−1xit−11 ); Left axis—probability of rejecting
the H0 on the 5% level and thus suggesting FE: gray shaded area == Hausman Test.
In the likely case that omitted trends are not common to all units (Table 3b),
period dummies can no longer guarantee the unbiasedness of the fixed-effects
model. In this case, the period fixed effects capture the mean of these
unit-specific trends so that residuals for other units, units that follow a
different trend, still show serial correlation, which of course can be
correlated with the variable of interest and, indeed, will almost certainly be
correlated if the variable of interest is also trended in a unit-specific
fashion. Our results thus run directly counter to Allan and Scruggs (Reference
Allan and Scruggs2004, 505) belief that “fixed effects do allow us to reduce the
possibility that the substantive estimates are in fact attributable to
country-specific trends.” We find this statement unlikely to be correct.
Instead, the presence of unit-specific trends that are not otherwise accounted
for renders the choice of a fixed-effects model more problematic.
In general, these Monte Carlo analyses provide ample evidence that the bias of
the fixed-effects model depends on the existence of dynamic misspecifications
and on the degree to which econometric solutions capture the dynamic
misspecification. The inclusion of the correct dynamic model provides of course
a solution but it is usually hard to test for the source of dynamic
misspecifications, especially when different dynamic issues occur jointly.
Different dynamic misspecifications can lead to similar manifestations in the
residuals, e.g. serial correlation. However, not every econometric model
controlling for autocorrelation (e.g. LDV, ADL, Prais–Winsten) will treat the
source of the problem successfully and might exacerbate the bias.
5.3 EXPERIMENT 3. MISSPECIFIED LAG STRUCTURE
In the final set of simulations we study the impact of a very common dynamic
misspecification (Adolph, Butler, and Wilson Reference Adolph, Butler and
Wilson2005; Wilson and Butler Reference Wilson and Butler2007) on the
performance of pooled-OLS and fixed-effects estimators. Many applied researchers
do not sufficiently explore the potential of lagged effects on the outcome.
Often, ignoring lagged effects will result in not rejecting the Null Hypothesis,
concluding that there are no effects from xx on yy (Plümper, Troeger, and Manow
Reference Plümper, Troeger and Manow2005). In models with several
right-hand-side variables and complex dynamics, especially when analyzing pooled
data, it becomes very difficult if not impossible to test for the correct lag
length of right-hand-side variables.
In pooled social science data we also find very often that effects are delayed
differently for different units. The lag length can vary because for example
different electoral systems generate different political reaction functions. It
is conceivable that changes in the political color of the executive have
differently delayed effects on political outcomes in coalition vs. single party
governments due to different bargaining situations (Plümper, Troeger, and Manow
Reference Plümper, Troeger and Manow2005). Table 4a presents the results for an
un-modeled (except in the ADL(1,1) specification) one period lagged effect of
x1itxit1 , while Table 4b presents MC findings for unit-specific lag length.
Table 4a. Misspecified lag of RHS variable: bias for estimate of x1itxit1 and
x1it−1xit−11 .
Note: Right axis—absolute bias: —— OLS, - - - - - FE, ⋯⋯⋯⋯ A–B (ADL: gray lines
== bias of coefficient for x1it−1xit−11 ); Left axis—probability of rejecting
the H0 on the 5% level and thus suggesting FE: gray shaded area == Hausman Test.
Table 4b. Misspecified unit-specific lag of RHS variable: Bias for Estimate of
x1itxit1 and x1it−1xit−11 .
Note: Right Axis—Absolute Bias: —— OLS, - - - - - FE, ⋯⋯⋯⋯ A–B (ADL: gray lines
== bias of coefficient for x1it−1xit−11 ); Left Axis—Probability of rejecting
the H0 on the 5% level and thus suggesting FE: gray shaded area == Hausman Test.
Experiment 3 adds further support to the notion that dynamic misspecification
biases fixed-effects estimates and that this bias can outweigh the bias of
pooled-OLS estimates facing the same dynamic problems. We also find evidence
that common econometric solutions to dynamic misspecifications can exacerbate
the bias. The results for experiment 4 are indeed staggering: All dynamic
specifications except the ADL(1,1) model produce largely biased estimates when
the correct lag length is ignored. The bias generated by including unit-specific
effects in these cases exceeds 100 per cent. This, in our perspective,
potentially provides the best argument for preferring pooled-OLS to the
fixed-effects model when dynamics are not explicitly modeled by substantive
variables or the correct dynamic specification but controlled away by
econometric patches. However, in the presence of dynamic misspecification,
neither fixed effects nor pooled-OLS will be unbiased.
Only econometric specifications that explicitly include a one period lagged
right-hand-side variable ( x1it−1xit−11 ) like the ADL (1,1) model can recover
the true effect of x1it−1xit−11 . In the simpler case where xx exerts a uniform
one period lagged effect on the outcome yy (Table 4a) both OLS and FE estimation
produces unbiased estimates of x1it−1xit−11 (gray lines) which is included in
the DGP. The FE estimator also generates unbiased estimates for x1itxit1 which
is an unnecessary element while the OLS estimator produces slightly biased
estimates of x1itxit1 (black lines). In the more complex situation where lag
structures are unit-specific (Table 4b), both estimators produce biased
estimates for x1it−1xit−11 (gray lines), and this problem appears to affect the
fixed-effects estimator more strongly than pooled-OLS. In comparison, either FE
or pooled-OLS are able to recover the zero effect of the unnecessary component
x1itxit1 (black lines), with the FE estimator performing slightly better
especially as TT grows larger.
The poor performance of the Hausman test is starkest in this set of experiments.
Independent of existing correlation between unit-specific effects and RHS
variables and independent of whether the FE model generates a larger bias than
an OLS or RE specification, the Hausman test indiscriminately and wrongly favors
the FE estimator.
The first best strategy to estimate models with complex dynamics in the true
data-generating process, heterogeneous lag structures, time-varying
conditionality, trended regressors, and so on, is to actually try modeling these
dynamics directly rather than eliminating serially correlated errors. This error
structure exists not because nature invented a complex error process that ought
to be controlled away, but because of a dynamic misspecification in the
underlying data-generating process. A fixed-effects model with some added fixes
for dynamics does not offer a valid strategy for analyzing dynamic phenomena in
the social sciences. Our findings for pooled data with relatively large TT are
consistent with recent research on short dynamic panels and correlated
unit-specific effects (Pickup Reference Pickup2017).
Our results also support findings by Adolph, Butler, and Wilson (Reference
Adolph, Butler and Wilson2005) as well as Wilson and Butler (Reference Wilson
and Butler2007) who have demonstrated that the use of so called dynamic panel
models (Arellano and Bond Reference Arellano and Bond1991; Blundell and Bond
Reference Blundell and Bond1998 etc.) does not alleviate bias from other dynamic
misspecifications, even simple ones but only the Nickell-bias that stems from
combining fixed effects with a lagged dependent variable. Even if the dynamics
in the data-generating process remain fairly trivial, we find substantive bias
in the Arellano–Bond model.
5.4 DISCUSSION
The MC analyses we conduct do not tackle the question whether social scientists
can manage to model dynamics properly. Widely used ‘from the shelf’ model
specifications such as the fixed-effects model with a lagged dependent variable,
with period fixed effects, or the Arellano–Bond model reveal substantive bias if
the data-generating process assumed in the simulations is not completely
trivial. Yet, true data-generating processes usually tend to be much more
complex than the ones we design here. We further demonstrate that the widely
employed fixed-effects estimator performs poorly, and often even worse than our
benchmark, the naïve pooled-OLS model, which is widely criticized for its poor
properties. We do not argue that our analyses rehabilitate the pooled-OLS model
because its poor performance in the presence of unobserved unit-specific effects
and other misspecifications is widely studied and known.
6 CONCLUSION
The fixed-effects estimator is consistent in the presence of omitted variables
with time-invariant effects. It is not consistent in the presence of dynamic
misspecification. The fixed-effects estimator deals with one problem and one
problem only: its consistency depends on the strong assumption of the strict
absence of any specification error other than omitted constant variables with
effects that are entirely independent of time. These conditions are not likely
to exist in real social sciences data, where few if any variables have constant
effects over time.
Dynamic misspecification does not merely render the fixed-effects model biased.
Instead we demonstrate in this article that the fixed-effects estimator
amplifies the bias from dynamic misspecification relative to estimators that do
not shelter the estimation from the between variation. The increase of bias from
dynamic misspecification potentially reaches the point where the combined bias
from omitted time-invariant variables and dynamic misspecification of OLS
estimates becomes smaller than the bias of the fixed-effects model from dynamic
misspecification alone.
One could feel tempted to argue that the fixed-effects model solves one
particular problem perfectly and thus advise to use the fixed-effects estimator
in the likely presence of this problem and deal with all other issues through
other model specifications. However, this solution would only be convincing if
researchers could eliminate all other model misspecification or if FE would not
influence the bias that emanates from model misspecifications which FE do not
treat. But as we have demonstrated: this latter assumption is wrong: the use of
the fixed-effects model can increase the bias from dynamic misspecifications
relative to the naïve pooled-OLS model. Therefore, the case for the FE estimator
is limited to situations in which researchers are confident and can thus
plausibly argue that they have gotten the dynamic specification of their
empirical model correct. Our analyses suggest that simple econometric solutions
for modeling dynamics are not very likely to guarantee a correct dynamic
specification.Footnote 28 Our results demonstrate the importance of carefully
modeling underlying dynamics before testing for the existence and potential
correlation of unit-specific time-invariant heterogeneity.
These results have rather general implications for econometric research:
Misspecifications of the empirical model are not necessarily additive so that
solving one problem does not strictly improve the overall performance of the
estimator. Quite the contrary is true: Model misspecifications interact with
each other so that accounting for one problem by an econometric solution may
actually exacerbate the overall bias and therefore increase the probability of
wrong inferences. Model misspecifications are not likely to be independent of
each other: empirical models suffer from numerous misspecifications (Box
Reference Box1976; Plümper, Troeger, and Manow Reference Plümper, Troeger and
Manow2005; Neumayer and Plümper Reference Neumayer and Plümper2017) and the
solution to one problem often renders another problem worse and more difficult
to solve. In other words, our analysis casts some doubt on the usefulness of the
econometric practice to ‘solve’ single model misspecifications in isolation. The
proof that estimators are consistent in respect to a single model
misspecification does not guarantee correct inferences if applied researchers
cannot plausibly guarantee that their empirical model suffers from the treated
misspecification alone.
SUPPLEMENTARY MATERIAL
For supplementary material accompanying this paper, please visit
https://doi.org/10.1017/pan.2018.17.
--------------------------------------------------------------------------------
FOOTNOTES
Authors’ note: We thank Jonathan Kropko and the participants of the workshop
“Modeling Politics & Policy in Time and Space” organized by Guy Whitten and
Scott Cook at Texas A&M for helpful comments and input.
The replication files for the MC analysis can be found on the PA dataverse:
Troeger and Pluemper (2017), “Replication Data for: Not so Harmless After All:
The Fixed-Effects Model”, doi:10.7910/DVN/RAUIHG, Harvard Dataverse.
Contributing Editor: Suzanna Linn
1 The good reputation the fixed-effects model enjoys among econometricians and
increasingly among applied researchers, is perhaps best summarized with the
following claim: “With panel data, always model the fixed effects using dummy
variables (…). Do not estimate random-effects models without ensuring that the
estimator is consistent with respect to the fixed-effects estimator (using a
Hausman test)” (Antonakis et al. Reference Antonakis, Bendahan, Jacquart and
Lalive2010, 1113). This quote demonstrates a common misperception of the Hausman
test (Hausman (Reference Hausman1978), see also Ahn and Low Reference Ahn and
Low1996; Frondel and Vance Reference Frondel and Vance2010). The Hausman test
does not test the consistency of the random-effects model, it tests whether the
random-effects model generates estimates that significantly differ from the
fixed-effects model. This would be an indirect test of the random-effects
model’s consistency if and only if omitted time-invariant variables were the
only reason that could produce such a significant difference in estimates. We
later demonstrate that with more than one model misspecification the Hausman
test does not reliably identify the less biased estimator.
2 Bell and Jones (Reference Bell and Jones2015) discuss the possibility of
different effect strengths for level and changes, which can also be interpreted
as dynamic misspecification (Bell and Jones Reference Bell and Jones2015). For a
discussion of fixed versus random effects see also Clark and Linzer (Reference
Clark and Linzerx2015). Note that for our specification of the data-generating
process in the Monte Carlo analyses, random effects and pooled-OLS give
identical point estimates and very similar standard errors. We therefore do not
report random-effects results but everything we say about pooled-OLS also
applies to random effects.
3 Note that we solely discuss omitting important dynamics such as trends or
time-varying variables or lags of RHS variables. We do not analyze the effect of
including unnecessary dynamics directly. However our MC analysis include an
element of including dynamic components that are not necessarily in the DGP. For
example, many fixes that are used to control for serial correlation in the error
term are not part of the DGP. Including a Lagged Dependent Variable (LDV) or
time fixed effects into the RHS of the model is an example for this. These
dynamic specifications may generate additional bias because these elements can
pick up variation that should be attributed to other elements in the DGP.
4 These statements are based on the assumption that within and between effects
are the same. If this is not the case it depends whether the researcher is
interested in between or within or average effects across time and space. We
discuss this issue in more detail later on.
5 Some authors (i.e. Gamm and Kousser Reference Gamm and Kousser2010)
demonstrate that their estimates are robust to a change from fixed-effects
estimates to pooled-OLS. In the light of our results, we believe this is a
useful research strategy.
6 The poor performance of the Hausman test for different misspecifications
including serial correlation, non-stationarity, and heteroscedasticity is known
(Arellano Reference Arellano1993; Ahn and Low Reference Ahn and Low1996; Bole
and Rebec Reference Bole and Rebec2013).
7 It also does not help that econometric textbooks usually do not define the
term ‘unobserved heterogeneity’, tend to be imprecise about the conditions under
which the fixed-effects estimator is consistent, and hardly ever discuss the
conditions under which the fixed-effects model generates biased and inconsistent
estimates—at least not in a way that non-econometricians understand easily
(Hendry Reference Hendry1995; Baltagi Reference Baltagi2001; Wooldridge
Reference Wooldridge2002; Hsiao Reference Hsiao2014). Interestingly,
identification textbooks discuss the FE model’s properties in greater detail,
see Angrist and Pischke (Reference Angrist and Pischke2009) and the excellent
discussion in Morgan and Winship (Reference Morgan and Winship2007).
8 Variables that are usually treated as time-invariant including culture (Kayser
and Satyanath Reference Kayser and Satyanath2014), distance (Wegener Reference
Wegener1912), institutions (North Reference North1990), genetic markers (Hedrick
Reference Hedrick2005), tend to vary at least slowly over time. The only truly
time-invariant variable is ‘inheritance’. However, still in this case the
effects of inherited factors are not likely to be constant over time. Park
(Reference Park2012) develops a procedure that allows testing the assumption
that unobserved heterogeneity is indeed time-invariant.
9 Pickup (Reference Pickup2017) suggests a general-to-specific approach to
dynamics for ‘short panels’ and argues that researchers should first find a
plausible dynamic specification before dealing with unobserved heterogeneity.
10 “Substantive theory, then, typically does not provide enough guidance for
precise dynamic specifications” (DeBoef and Keele Reference DeBoef and
Keele2008, 196).
11 When political scientists employ distributed lag or error correction models,
they often do not include unit dummies, and when they use fixed effects, they
rarely control for complex dynamics. Exceptions exist, e.g. Haber and Menaldo
(Reference Haber and Menaldo2011) and Treisman (Reference Treisman2015) combine
unit and period fixed effects with an error correction model.
12 We do not wish to suggest here that error correction models and distributed
lag models allow social scientists to model dynamics correctly. These models do
assume homogeneous dynamic processes which neither capture omitted time-varying
variables nor unobserved time-varying conditionality of the variable of
interest, they remain limited in their ability to capture functional forms of
effects which do not simply diminish at a constant rate, and they de facto rely
on homogeneous lag structures.
13 The absence of theoretical guidance may be caused by theories which
“typically tell us only generally how inputs relate to processes we care about.
They are nearly always silent on which lags matter, (…), what characterizes
equilibrium behavior, or what effects are likely to be biggest in the long run”
(DeBoef and Keele Reference DeBoef and Keele2008, 186).
14 Fixed-effects estimates are also biased by what is known as the incidental
parameter problem (Neyman and Scott Reference Neyman and Scott1948; Lancester
Reference Lancester2000; Hahn and Kuersteiner Reference Hahn and
Kuersteiner2011). This incidental parameter problem for fixed-effects estimation
of pooled data is insofar interesting for our argument because it implies that
the fixed-effects estimator is consistent when TT approaches infinity. From this
perspective using fixed effects becomes a catch 22 because as the number of
periods increases, fixed-effects estimates become more precise but the
probability of dynamic misspecification bias increases as well.
15 If bias exclusively results from correlation of the between variation of
xitxit and witwit , it is of course possible to throw away all the between
variation and regress the y¨y¨ on x¨x¨ —the within variation of y on the within
variation of x—for an unbiased estimate.
16 We could add here the analysis of the random-effects model but this should
produce exactly the same average bias as an OLS model.
17 The replication files for the MC analysis can be found on the PA dataverse:
Troeger, Vera; Pluemper, Thomas, 2017, “Replication Data for: Not so Harmless
After All: The Fixed-Effects Model”, doi:10.7910/DVN/RAUIHG, Harvard Dataverse.
18 See Acemoglu et al. (Reference Acemoglu, Johnson, James, Robinson and
Yared2008) for the choice of an Arellano–Bond model, Beck and Katz (Reference
Beck and Katz1995) for the use of the lagged dependent variable (but see Achen
Reference Achen2000 and Keele and Kelly Reference Keele and Kelly2006), Huber
and Stevens (2012) for the Prais–Winsten transformation, and Becker and
Woessmann (Reference Becker and Woessmann2013) for the inclusion of period
dummies. For a broader discussion see DeBoef and Keele (Reference DeBoef and
Keele2008).
19 Since the combination of a LDV and unit-specific effects generates
Nickell-bias (Nickell Reference Nickell1981) we also show results for the most
common solution to Nickell-bias—and Arellano–Bond model (Arellano and Bond
Reference Arellano and Bond1991).
20 We have conducted additional experiments. Since findings remain consistent
with the results discussed here, we do not report additional findings.
21 In experiments not shown here we also studied bias of the fixed-effects model
with a binary treatment variable (Beck and Katz Reference Beck and Katz2001;
Green, Kim, and Yoon Reference Green, Kim and Yoon2001). Binary treatments can
be trended if the probability of treatment increases or declines over time.
Epidemics may serve as the most obvious example. Furthermore, most studies of
treatment effects only observe two periods: pre-treatment and post-treatment. In
this situation, the probability of treatment increases from zero to a
probability determined by the share of the treated cases to the total cases. In
such a case, every omitted trended variable will bias the results unless the
effect of this variable is strictly identical for treatment and control group.
Because of limited space we relegate the results for binary treatment variables
to an online appendix.
22 This set-up might seem somewhat unrealistic but we run the same experiment
with one half of the units positively trended and one half not trended and get
similar results.
23 Since the combination of unit fixed effects and a lagged dependent variable
induces additional bias, the so called Nickell-bias (Nickell Reference
Nickell1981), we also run dynamic panel models that allow for the combination of
unit-specific effects and a lagged dependent variable.
24 The OLS variant with Prais–Winsten transformation results in a GLS model.
25 Note that generalization from asymptotic properties to small sample
properties are not valid. At the same time, this logic overlooks multiple other
reasons for parameter heterogeneity.
26 We show detailed results for the bias of the coefficient of the LDV for all
MC experiment in Appendix tables A3 to A9.
27 In some cases the dotted line for the bias of the A–B estimator cannot be
seen because it is equal to the bias produced by the FE estimator and they
completely overlap.
28 The Fixed-Effects Estimator is of course the correct choice if researchers
are theoretically and empirically only interested in within effects. In this
case the fixed-effects estimator will give a more adequate econometric answer,
though it will still suffer from bias induced by dynamic misspecifications.
Throughout this paper we have assumed that within and between effects are the
same. This assumption is essential for our conclusions because only if it is
met, using between variation in addition to within variation to identify the
effects will generate less biased and more reliable estimates. However, as
mentioned before, we are not advocating using OLS over FE but are using OLS
estimates as benchmark because the undesirable properties are known in the
presence of misspecifications.
--------------------------------------------------------------------------------
REFERENCES
Acemoglu, D., Johnson, S., James, A., Robinson, J. A., and Yared, P.. 2008.
Income and democracy. American Economic Review 98(3):808–842.Google Scholar
Achen, C. H.2000. Why lagged dependent variables can suppress the explanatory
power of other independent variables. Presented at the Annual Meeting of the
Political Methodology, Los Angeles.Google Scholar
Adolph, C., Butler, D. M., and Wilson, S. E.. 2005. Like shoes and shirt, one
size does not fit all: Evidence on time series cross-section estimators and
specifications from Monte Carlo experiments. unpubl. Manuscript, Harvard
University.Google Scholar
Ahn, S. C., and Low, S.. 1996. A reformulation of the Hausman-test for
regression models with pooled cross-section-time-series data. Journal of
Econometrics 71:309–319.Google Scholar
Ahn, S. C., Lee, Y. H., and Schmidt, P.. 2013. Panel data models with multiple
time-varying individual effects. Journal of Econometrics 174:1–14.Google Scholar
Allan, J. P., and Scruggs, L.. 2004. Political partisanship and welfare state
reform in advanced industrial societies. American Journal of Political Science
48(3):496–512.Google Scholar
Angrist, J. D., and Pischke, J. S.. 2009. Mostly harmless econometrics. An
empiricists’ companion . Princeton: Princeton University Press.Google Scholar
Antonakis, J., Bendahan, S., Jacquart, P., and Lalive, R.. 2010. On making
causal claims: a review and recommendations. Leadership Quarterly
21:1086–1120.Google Scholar
Arellano, M. 1993. On the testing of correlated effects with panel data. Journal
of Econometrics 59(1–2):87–97.Google Scholar
Arellano, M., and Bond, S.. 1991. Some tests of specification for panel data:
Monte Carlo evidence and an application to employment equations. Review of
Economic Studies 58:277–297.Google Scholar
Baltagi, B. 2001. Econometric analysis of panel data . John Wiley & Sons.Google
Scholar
Beck, N., and Katz, J. N.. 1995. What to do (and not to do) with time-series
cross-section data. American Political Science Review 89(03):634–647.Google
Scholar
Beck, N., and Katz, J. N.. 2001. Throwing out the Baby with the Bath Water: A
comment on Green, Kim, and Yoon. International Organization 55:487-+.Google
Scholar
Becker, S. O., and Woessmann, L.. 2013. Not the opium of the people: Income and
secularization in a panel of Prussian counties. American Economic Review
103(3):539–544.Google Scholar
Bell, A., and Jones, K.. 2015. Explaining fixed effects: Random effects modeling
of time-series cross-sectional and panel data. Political Science Research and
Methods 3(01):133–153.Google Scholar
Besley, T., and Reynal-Querol, M.. 2011. Do democracies select more educated
leaders? American Political Science Review 105(3):552–566.Google Scholar
Blundell, R., and Bond, S.. 1998. Initial conditions and moment restrictions in
dynamic panel data models. Journal of Econometrics 87(1):115–143.Google Scholar
Bole, V., and Rebec, P.. 2013. Bootstrapping the Hausman test in panel data
models. Communications in Statistics-Simulation and Computation
42(3):650–670.Google Scholar
Box, G. E. P. 1976. Science and Statistics. Journal of the American Statistical
Association 71:791–799.Google Scholar
Clark, T. S., and Linzerx, D. A.. 2015. Should I use fixed or random effects?
Political Science Research and Methods 3(02):399–408.Google Scholar
DeBoef, S., and Keele, L. J.. 2008. Taking time seriously: Dynamic regression.
American Journal of Political Science 52(1):184–200.Google Scholar
Egorov, G., Guriev, S., and Sonin, K.. 2009. Why resource-poor dictators allow
freer media: A theory and evidence from panel data. American Political Science
Review 103(4):645–668.Google Scholar
Franzese, R. J. Jr. 2003a. Multiple hands on the wheel: empirically modeling
partial delegation and shared policy control in the open and institutionalized
economy. Political Analysis 445–474.Google Scholar
Franzese, R.J. 2003b. Quantitative empirical methods and the
context-conditionality of classic and modern comparative politics. CP:
Newsletter of the Comparative Politics Organized Section of the American
Political Science Association 14(1):20–24.Google Scholar
Franzese, R. J. Jr, and Hays, J. C.. 2007. Spatial econometric models of
cross-sectional interdependence in political science panel and
time-series-cross-section data. Political Analysis 140–164.Google Scholar
Franzese, R., and Kam, C.. 2009. Modeling and interpreting interactive
hypotheses in regression analysis . University of Michigan Press.Google Scholar
Frondel, M., and Vance, C.. 2010. Fixed, random, or something in between? A
variant of Hausman’s specification test for panel data estimators. Economics
Letters 107:327–329.Google Scholar
Gamm, G., and Kousser, T.. 2010. Broad bills or particularistic policy?
Historical patterns in American state legislatures. American Political Science
Review 104(1):151–170.Google Scholar
Gerber, A. S., Gimpel, J. G., Green, D. P., and Shaw, D. R.. 2011. How large and
long-lasting are the persuasive effects of televised campaign ads? Results from
a randomized field experiment. American Political Science Review
105(01):135–150.Google Scholar
Getmansky, A., and Zeitzoff, T.. 2014. Terrorism and voting: The effect of
rocket threat on voting in Israeli elections. American Political Science Review
108(3):588–604.Google Scholar
Godfrey, L. G. 1998. Hausman tests for autocorrelation in the presence of lagged
dependent variables Some further results. Journal of econometrics
82(2):197–207.Google Scholar
Green, D. P., Kim, S. Y., and Yoon, D. H.. 2001. Dirty Pool. International
Organization 55:441-+.Google Scholar
Guisinger, A., and Singer, D. A.. 2010. Exchange Rate proclamations and
inflation-fighting credibility. International Organization 64(2):313–337.Google
Scholar
Haber, S., and Menaldo, V.. 2011. Do natural resources fuel authoritarianism? A
reappraisal of the resource curse. American Political Science Review
105(01):1–26.Google Scholar
Hahn, J., and Kuersteiner, G.. 2011. Bias reduction for dynamic nonlinear panel
models with fixed effects. Econometric Theory 27:1152–1191.Google Scholar
Harris, M. N., Kostenko, W., Matyas, L., and Timol, I.. 2009. The robustness of
estimators for dynamic panel data models to misspecification. Singapore Economic
Review 54:399–426.Google Scholar
Hausman, J. A. 1978. Specification tests in econometrics. Econometrica
46(6):1251–1271.Google Scholar
Hedrick, P. W. 2005. A standardized genetic differentiation measure. Evolution
59:1633–1638.Google Scholar
Hendry, D. F. 1995. Dynamic econometrics . Oxford: Oxford University
Press.Google Scholar
Hoechle, D. 2007. Robust standard errors for panel regressions with
cross-sectional dependence. Stata Journal 7(3):281.Google Scholar
Hsiao, C. 2014. Analysis of panel data . Cambridge: Cambridge University
Press.Google Scholar
Huber, J. D., and Stevens, E.. 2012. Democracy and the left: Social policy and
inequality in Latin America. Journal of Social Policy 42(3):660–661.Google
Scholar
Humphreys, M., and Weinstein, J. M.. 2006. Handling and manhandling civilians in
civil war. American Political Science Review 100(3):429.Google Scholar
Kayser, M. A., and Satyanath, S.. 2014. Fairytale Growth. Unp. Manuscript,
Hertie School of Governance, Berlin.Google Scholar
Kayser, M. A. 2009. Partisan waves: International business cycles and electoral
choice. American Journal of Political Science 53(4):950–970.Google Scholar
Keele, L. J., and Kelly, N. J.. 2006. Dynamic models for dynamic theories: The
ins and outs of LDVs. Political Analysis 14(2):186–205.Google Scholar
Keele, L., Linn, S., and Webb, C. M.. 2016. Treating time with all due
seriousness. Political Analysis 24(1):31–41.Google Scholar
Kiviet, J. F. 1995. On bias, inconsistency, and efficiency of various estimators
in dynamic panel data models. Journal of Econometrics 68(1):53–78.Google Scholar
Kogan, V., Lavertu, S., and Peskowitz, Z.. 2016. Performance federalism and
local democracy: Theory and evidence from school tax referenda. American Journal
of Political Science 60(2):418–435.Google Scholar
Lancester, T. 2000. The Incidental Parameter Problem since 1948. Journal of
Econometrics 95:391–413.Google Scholar
Lebo, M. J., McGlynn, A. J., and Koger, G.. 2007. Strategic party government:
Party influence in congress, 1789–2000. American Journal of Political Science
51(3):464–481.Google Scholar
Lee, Y. 2012. Bias in dynamic panel models under time series misspecification.
Journal of Econometrics 169:54–60.Google Scholar
Lipsmeyer, C. S., and Zhu, L.. 2011. Immigration, globalization, and
unemployment benefits in developed EU states. American Journal of Political
Science 55(3):647–664.Google Scholar
Lupu, N., and Pontusson, J.. 2011. The structure of inequality and the politics
of redistribution. American Political Science Review 105(02):316–336.Google
Scholar
Menaldo, V. 2012. The middle east and north Africa’s resilient monarchs. The
Journal of Politics 74(3):707–722.Google Scholar
Morgan, S. L., and Winship, C.. 2007. Counterfactuals and causal inference.
Methods and principles for social research . Cambridge: Cambridge University
Press.Google Scholar
Mukherjee, B., Smith, D. L., and Li, Q.. 2009. Labor (im) mobility and the
politics of trade protection in majoritarian democracies. Journal of Politics
71(1):291–308.Google Scholar
Neumayer, E., and Plümper, T.. 2017. Robustness tests for quantitative research
. Cambridge University Press.Google Scholar
Neumayer, E., and Plümper, T.. 2016. W. Political Science Research and Methods
4(1):175–193.Google Scholar
Neyman, J., and Scott, E.. 1948. Consistent estimates based on partially
consistent observations. Econometrica 16:1–32.Google Scholar
Nickell, S. 1981. Biases in dynamic models with fixed effects. Econometrica
49:1417–1426.Google Scholar
North, D. C. 1990. Institutions, institutional change, and economic performance
. Cambridge: Cambridge University Press.Google Scholar
Park, J. H. 2012. A unified method for dynamic and cross-sectional
heterogeneity: Introducing hidden markov panel models. American Political
Science Review 56:1040–1054.Google Scholar
Pickup, M.2017. A general-to-specific approach to dynamic panel models with a
very small TT . Presented to the 2017 Meeting of the Midwest Political Science
Association, Chicago Illinois.Google Scholar
Plümper, T., Troeger, V. E., and Manow, P.. 2005. Panel data analysis in
comparative politics: Linking method to theory. European Journal of Political
Research 44(2):327–354.Google Scholar
Plümper, T., and Troeger, V. E.. 2007. Efficient estimation of time-invariant
and rarely changing variables in finite sample panel analyses with unit fixed
effects. Political Analysis 15(2):124–139.Google Scholar
Plümper, T., and Troeger, V. E.. 2011. Fixed-effects vector decomposition:
properties, reliability, and instruments. Political Analysis
19(2):147–164.Google Scholar
Ross, M. L. 2008. Oil, Islam, and women. American Political Science Review
102(1):107–123.Google Scholar
Soroka, S. N., Stecula, D. A., and Wlezien, C.. 2015. It’s (change in) the
(future) economy, stupid: economic indicators, the media, and public opinion.
American Journal of Political Science 59(2):457–474.Google Scholar
Treisman, D. 2015. Income, democracy, and leader turnover. American Journal of
Political Science 59(4):927–942.Google Scholar
Troeger, Vera, and Pluemper, Thomas. 2017. Replication Data for: Not so Harmless
After All: The Fixed-Effects Model. doi:10.7910/DVN/RAUIHG, Harvard
Dataverse.Google Scholar
Wegener, A. 1912. Die Entstehung der Kontinente. Geologische Rundschau
3:276–292.Google Scholar
Wilson, S. E., and Butler, D. M.. 2007. A lot more to do: The sensitivity of
time-series cross-section analyses to simple alternative specifications.
Political Analysis 15(2):101–123.Google Scholar
Wooldridge, J. M. 2002. Econometric analysis of cross section and panel data .
Cambridge, MA: MIT Press.Google Scholar
View in content
Table 1. Bias over all Experiments.
--------------------------------------------------------------------------------
View in content
Table 2. Omitted Within Variance corr(x¨1it,x¨2it)=0.5corr(x¨it1,x¨it2)=0.5:
Bias for Estimate of x1itxit1 and x1it−1xit−11.
--------------------------------------------------------------------------------
View in content
Table 3a. Omitted common trend: bias for estimate of x1itxit1 and x1it−1xit−11.
--------------------------------------------------------------------------------
View in content
Table 3b. Omitted unit-specific trends: bias for Estimate of x1itxit1 and
x1it−1xit−11.
--------------------------------------------------------------------------------
View in content
Table 4a. Misspecified lag of RHS variable: bias for estimate of x1itxit1 and
x1it−1xit−11.
--------------------------------------------------------------------------------
View in content
Table 4b. Misspecified unit-specific lag of RHS variable: Bias for Estimate of
x1itxit1 and x1it−1xit−11.
PLÜMPER AND TROEGER SUPPLEMENTARY MATERIAL
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VARYING RESPONSES TO COMMON SHOCKS AND COMPLEX CROSS-SECTIONAL DEPENDENCE:
DYNAMIC MULTILEVEL MODELING WITH MULTIFACTOR ERROR STRUCTURES FOR TIME-SERIES
CROSS-SECTIONAL DATA
Type Article Title Varying Responses to Common Shocks and Complex
Cross-Sectional Dependence: Dynamic Multilevel Modeling with Multifactor Error
Structures for Time-Series Cross-Sectional Data Authors Xun Pang Journal
Political Analysis
Published online: 4 January 2017
REGRESSION TECHNIQUES FOR INTEGRATED FINANCIAL TIME SERIES
Type Chapter Title Regression techniques for integrated financial time series
Authors Terence C. Mills and Raphael N. Markellos Journal The Econometric
Modelling of Financial Time Series
Published online: 5 June 2012
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--------------------------------------------------------------------------------
Hou, Yue and Li, Siyao 2020. How Local Leadership Rotation Breaks State-Business
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* CrossRef
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--------------------------------------------------------------------------------
Казаринова, Дарья 2020. “Массовая наука” на конференции ЕPSA: опыт включенного
наблюдения. Полис. Политические исследования, p. 163.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Lewallen, Jonathan 2020. Booster seats: new committee chairs and legislative
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* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Savage, Jesse Dillon 2020. Political Survival and Sovereignty in International
Relations.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
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30, Issue. 10, p. 2487.
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--------------------------------------------------------------------------------
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for European integration, 1995–2018. European Union Politics, Vol. 22, Issue. 2,
p. 266.
* CrossRef
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--------------------------------------------------------------------------------
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Rule of Law and Its Social Reception as Determinants of Economic Development: A
Comparative Analysis of Germany and Poland. Law and Development Review, Vol. 14,
Issue. 2, p. 359.
* CrossRef
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--------------------------------------------------------------------------------
Hoover, Joe Atari, Mohammad Mostafazadeh Davani, Aida Kennedy, Brendan
Portillo-Wightman, Gwenyth Yeh, Leigh and Dehghani, Morteza 2021. Investigating
the role of group-based morality in extreme behavioral expressions of prejudice.
Nature Communications, Vol. 12, Issue. 1,
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Zohlnhöfer, Reimut and Voigt, Linda 2021. The partisan politics of employment
protection legislation: Social democrats, Christian democrats, and the
conditioning effect of unemployment. European Political Science Review, Vol. 13,
Issue. 3, p. 331.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Yeung, Eddy SF 2021. Does immigration boost public Euroscepticism in European
Union member states?. European Union Politics, Vol. 22, Issue. 4, p. 631.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Fedotenkov, Igor and Idrisov, Georgy 2021. A supply-demand model of public
sector size. Economic Systems, Vol. 45, Issue. 2, p. 100869.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
JACOBS, ALAN M. MATTHEWS, J. SCOTT HICKS, TIMOTHY and MERKLEY, ERIC 2021. Whose
News? Class-Biased Economic Reporting in the United States. American Political
Science Review, Vol. 115, Issue. 3, p. 1016.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
ENGLER, FABIAN 2021. Political parties, globalization and labour strength:
Assessing differences across welfare state programs. European Journal of
Political Research, Vol. 60, Issue. 3, p. 670.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Schmitt, Carina 2022. Handbuch Policy-Forschung. p. 1.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Goubin, Silke and Kumlin, Staffan 2022. Political Trust and Policy Demand in
Changing Welfare States: Building Normative Support and Easing Reform
Acceptance?. European Sociological Review, Vol. 38, Issue. 4, p. 590.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Grasse, Nathan J Searing, Elizabeth A M and Neely, Daniel G 2022. Finding Your
Crowd: The Role of Government Level and Charity Type in Revenue Crowd-Out.
Journal of Public Administration Research and Theory, Vol. 32, Issue. 1, p. 200.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Lindberg, Staffan I. Lo Bue, Maria C. and Sen, Kunal 2022. Clientelism,
corruption and the rule of law. World Development, Vol. 158, Issue. , p. 105989.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Thombs, Ryan P. 2022. A Guide to Analyzing Large N, Large T Panel Data. Socius:
Sociological Research for a Dynamic World, Vol. 8, Issue. , p. 237802312211176.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Pickup, Mark and Hopkins, Vincent 2022. Transformed-likelihood estimators for
dynamic panel models with a very small T. Political Science Research and
Methods, Vol. 10, Issue. 2, p. 333.
* CrossRef
* Google Scholar
--------------------------------------------------------------------------------
Limberg, Julian 2022. Building a tax state in the 21st century: Fiscal pressure,
political regimes, and consumption taxation. World Development, Vol. 154, Issue.
, p. 105879.
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--------------------------------------------------------------------------------
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