The Estimation of Time-Invariant Variables
in Panel Analyses with Unit Fixed Effects
THOMAS PLÜMPER#
#
AND VERA E. TROEGER
+
University of Konstanz, Department of Political Science and Public
Administration, Box D 86; D-78457 Konstanz, Germany; Tel: +49-7531-88
2608/3081; Fax: +49-7531-88 2774; E-Mail: Thomas.Pluemper@unikonstanz.de
+
Max-Planck-Institute for Research into Economic Systems, Kahlaische Strasse
10, D-07745 Jena, Germany
Summary: This paper analyzes the estimation of time-invariant variables in panel
data models with unit-effects. We compare three procedures that have frequently
been employed in comparative politics, namely pooled-OLS, random effects and
the Hausman-Taylor model, to a vector decomposition procedure that allows
estimating time-invariant variables in an augmented fixed effects approach. The
procedure we suggest consists of three stages: the first stage runs a fixed-effects
model without time-invariant variables, the second stage decomposes the uniteffects vector into a part explained by the time-invariant variables and an error
term, and the third stage re-estimates the first stage by pooled-OLS including the
time invariant variables plus the error term of stage 2. We use Monte Carlo
simulations to demonstrate that this method works better than its alternatives in
estimating typical models in comparative politics. Specifically, the unit fixed
effects vector decomposition technique performs better than both pooled OLS and
random effects in the estimation of time-invariant variables correlated with the
unit effects and better than Hausman-Taylor in estimating the time-invariant
variables correlated with the unit effects. Finally, we re-analyze recent work by
Huber and Stephens (2001) as well as by Beramendi and Cusack (2004). These
analyses seek to cope with the problem of time-invariant variables in panel data.
John Stephens and Pablo Beramendi provided not only access to their data but
also assisted our replication attempts. We thank Neal Beck, Christian Kraft,
Richard Stazinski for helpful comments. The usual disclaimer applies.
2
The Estimation of Time-Invariant Variables
in Panel Analyses with Unit Fixed Effects
1.
Introduction
Since the publication of Nathaniel Beck’s and Jonathan Katz’s influential article
on pooled time-series analysis (Beck/ Katz 1995) analyzing panel data has
become a standard in comparative political science. As political scientists have
frequently observed, standards provide focal points and they are applied because
deviations from these standards are likely to be costly. In comparative politics,
referees may impose ‘sanctions’ on researchers that deviate from pooling. Beyond
this obvious incentive structure, pooling also has some objective advantages.
Social scientists are often interested in explaining the variance of trends across
units so that neither cross-sectional nor time-series analyses provide relevant tests
of the theory. Hence, pooling is warranted. At times, researchers have analyzed
panel data because pooling increases the number of observations.
The most important methodical advantage of panel data relates to the growing
concern of applied researchers about bias resulting from omitted variables.
Contrary to a cross-section or a pure time-series panel data analyses allow
controlling for unit fixed-effects that – as most researchers believe – capture the
systematic influences from omitted variables. This belief is not wrong but
potentially misleading, since unit fixed effects do not eliminate all kinds of
omitted variable bias. Time-variant omitted variables may still bias the estimates.
Thus, one danger of fixed effects models is that many researchers believe that the
inclusion of unit dummies precludes problems with omitted variables.
Unit effects represent “all factors (…) that do not change over time” (Wooldridge
1999: 420). In other words, unit-fixed effects account for time-invariant crosssectional effects – may they be observed or unobserved. This has two obvious but
notable consequences: on the one hand, unit dummies do not necessarily eliminate
omitted variable bias; on the other hand, unit fixed effects models cannot estimate
the coefficients of theoretically interesting time-invariant variables.
For the latter reason the quest to include or exclude country dummies has
stimulated a lively debate among political scientists. While some authors suggest
that country dummies are needed to account for the “underlying historical fabric
(…) that is not captured by any of the time and country-varying regressors”
(Garrett/ Mitchell 2001: 163), others claim that unit fixed effects throw out the
3
baby with the bath water”, because political scientist are mainly interested in
institutional effects and institutions do not vary much over time (Kittel/ Obinger
2002: 21).
This paper discusses potential remedies for the estimation of time-invariant
variables in panel-data analyses with unit effects. We compare three procedures
that have frequently been used in comparative politics – namely pooled-OLS,
random effects and the Hausman-Taylor formulation – to a vector decomposition
procedure that allows estimating time-invariant variables in an augmented fixed
effects approach. The model we advocate consists of three stages: in the first stage
we run a fixed-effects model, in the second stage the unit-effects vector is
decomposed into a part explained by the time-invariant variables and an error
term, in the third stage we re-estimate the first stage including the time invariant
variables and the error term obtained in stage 2 by pooled-OLS.
We evaluate the small-sample properties of the procedure we advocate (dubbed
xtfevd for fixed effect vector decomposition) to pooled OLS, random effects and
Hausman-Taylor. Monte Carlo simulations demonstrate that xtfevd is on average
less biased than the available alternatives. Moreover, xtfevd has better small
sample properties than Hausman-Taylor. At the same time it is less biased than
random effects and pooled OLS when unit effects are correlated with time-variant
variables. xtfevd outperforms pooled OLS and random effects in estimating timevariant variables that are correlated with the unit effects ( ui ) and it is more
adequate than Hausman-Taylor in calculating the effect of time-variant variables
that are correlated with the unit effects. Thus, employing the unit fixed effects
vector decomposition technique appears to be superior for the analysis of datagenerating processes typical for panel data in comparative politics.
We will also identify one situation in which neither of the four models does
particularly well. When the unit-effects are correlated with exogenous variables
and highly skewed, all methods give biased estimates.
The remainder of the paper is organized as follows. The next section briefly (re-)
introduces the alternative procedures to the reader. Section 3 presents the xtfevd
estimator. In section 4, we compare existing approaches by conducting Monte
Carlo simulations. Section 5 re-analyzes Evelyn Huber and John D. Stephens
(2001) examination of the impact of constitutional features on the dynamics of
social welfare spending and Pablo Beramendi and Thomas Cusack’s (2004)
analysis of the political determinants of income inequality. Section 6 concludes.
4
2. The estimation of time-invariant variables
Time-invariant variables can be distinguished into two broadly defined categories.
The first type subsumes variables that are time-invariant by definition. Often,
these variables measure geography or inheritance. Switzerland and Hungary are
both landlocked countries, they are both located in Central Europe, and there is
little nature and (hopefully) politics will do about it for the foreseeable future.
Along similar lines, a country may or may not have a colonial heritage or a
climate prone to tropical diseases.
The second category covers variables that are time-invariant for the period under
analysis or because of researchers’ selection of cases. For instance, constitutions
in postwar OECD countries have proven to be highly durable. Switzerland is a
democracy since 1291 and the US maintained a presidential system since
independence day. Yet, by increasing the number of periods and/or the number of
cases it would be possible to render these variables time-variant. This suggests
that many variables are time-invariant because of researchers’ deliberate choice.
At times, researchers even suppress time-varying information and operationalize a
theoretically plausible influence as (time-invariant) dichotomous variable even
though a (time-varying) continuous alternative exists.
Adding observations may be preferable to searching for remedies of estimating
panel data that include time-invariant variables. We suggest that researchers first
consider broadening their sample of units or lengthening the time period under
observation rather than employing one of the procedures we discuss here.
Nevertheless, all methods enable researchers to estimate variables that are timeinvariant by nature or by researchers’ decision.
One reason for suppressing time-series variance could be that variables rarely
change. The identification of the effect of an almost time-invariant variable is
difficult. By taking the unit means one can treat any changing variable as timeinvariant. Whether this is an appropriate research strategy clearly depends on the
alternatives. From our perspective it is likely that testing the theory within a
broader research design – one that includes more observations – is a superior
research strategy.
Most applied researchers have at one point been confronted with the appropriate
estimation of time-invariant variables of theoretical interest. Even a brief look into
datasets frequently used in political science reveals that time-invariant variables
5
are by no means rare. Many datasets contain variables that are time-invariant over
relatively long periods. For example, data on political institutions usually includes
constitutional variables that rarely and slowly change.
Since no standard method exists, astonishingly diverse ways to deal with the
problem of time-invariant variables have been employed in the literature. This
section briefly discusses the procedures in use. Before doing so, however, a few
remarks on notation are in order. We assume the following data generating
process
yit = α + X it β + Zi γ + u i + εit
(1)
where X it β stands for a vector of time-variant variables and Zi γ for a vector of
time-invariant variables, u i denotes the unit effect and εit is the normal
distributed error component. We further hold that u i is correlated with at least one
of the time-variant variables and at least one of the time-invariant variables:
p lim
N →∞
1 '
1
x1it u i = 0; p lim x '2it u i ≠ 0
N
N →∞ N
(2)
1 '
1
z1i u i = 0; p lim z '2i u i ≠ 0
N
N →∞ N
(3)
and
p lim
N →∞
These assumptions are likely to be satisfied in most data analyzed in comparative
politics.
Having said this, we now explain how applied researchers have dealt with the
estimation problems associated with this data generating process. To begin, all
three procedures have in common that researchers shied away from controlling for
fixed effects because the unit effect dummies and the time-invariant variables are
perfectly collinear. Scholars either run random effects (often regardless of
inconsistency), employ the Hausman–Taylor formulation (which is available in
Stata since version 8.0), or neglect individual effects altogether and rather run a
pooled OLS model. We discuss all potential remedies in turn.
The first ‘solution’ to the estimation problems posed by time-invariant variables is
to ignore the possibility of unit effects.1 For instance, Daron Acemoglu et al.
(2002) justify not controlling for unit effects by stating the following: “Recall that
1
Oaxaca and Geisler (2003) suggest a two-stage GLS estimator which is
equivalent to pooled OLS, but gives different standard errors.
6
our interest is in the historically-determined component of institutions (that is
more clearly exogenous), hence not in the variations in institutions from year-toyear. As a result, this regression does not (cannot) control for a full set of country
dummies.” (Acemoglu et al. 2002: 27)
Doing nothing about unit effects and running pooled OLS does not offer a way to
reduce omitted variable bias. As is well understood, the exclusion of exogenous
variables that influence the endogenous variable does cause bias, unless the
excluded variables and the included exogenous variables are orthogonal to each
other. The same holds true for excluded unit fixed effects. The exclusion of a
battery of unit dummies that captures ui biases estimates if ui is correlated with
both, yit and xit . Therefore, it seems unlikely to obtain unbiased coefficients by
pooled OLS, especially if N is small.
A strategy that in many situations should work more appropriate than pooled OLS
in the presence of time-invariant variables is adapting the random effects model.
Random effects should perform better than pooled OLS even if the Hausman test
suggests (Hausman 1978) that random effects are inconsistent and a fixed effects
specification is required. When time invariant variables preclude the estimation of
unit fixed effects, random effects may serve as a viable second best option.
One should keep in mind, however, that when the Hausman test rejects a random
effects specification, the procedure is inconsistent and very likely to be biased,
because random effects impose strict exogeneity of x it and orthogonality between
x it and u i :2
E ( u it | x i , u i ) = 0; t = 1,..., T
E ( x i | u i ) = E(u i ) = 0; x i = ( x i1 , x i2 ,..., x iT )
(5)
Moreover, random effects are based on a feasible gls estimator where the Omega
matrix has a special random effects structure. Rather than depending on T(T+1)/2
unrestricted variances and covariances as it is the case in a normal GLS model,
Omega only depends on the variances of u i and εit regardless of the size of T:
2
But probably most importantly, real world data rarely satisfies the conditions
under which random effects estimators are consistent. “For studies in political
science using TSCS data it will almost always be the case that the unobserved
local factors are captured by u i and correlated with X; indeed, the main reason to
abandon the standard pooled OLS is because we think such a correlation is likely
to exist.” (Wilson/ Butler 2003: 8) At least under this condition, random effects
models are second best options of a potentially dubious quality. We will later see
that random effects models do not work significantly better than pooled OLS.
7
Ω ≡ E ( v i vi ' )
vi = u i jT + εi ; jT is the T × 1vector of ones
(6)
Even if the data satisfies the random effects assumptions of strict exogeneity and
orthogonality between x it and u i , random effects models share the poor small
sample properties of GLS. As a consequence, one should expect that the random
effects procedure gives biased and inefficient estimates of the true betas in
relatively small samples.
To overcome the problem of random effects inconsistency, Hausman and Taylor
(1981) advocated the use of instruments for the variables that are likely to be
correlated with the random effects. Unfortunately, this correlation is unobservable
and thus it requires some imagination to correctly specify the Hausman-Taylor
model. This is rarely a trivial problem. The estimated coefficients largely vary
with researchers’ decision which variables are endogenous and which variables
are exogenous to the random effects. Hence, the Hausman-Taylor procedure
leaves researchers with a discretionary choice that largely influences the results.
As a straightforward remedy for the apparent problem of choosing a set of
instruments in the presence of an unknown correlation between the right-hand
side variables and the random effects, Hausman and Taylor suggest using the
exogenous variables that vary over time and are not correlated with the individualspecific part of the error term u i (x1) to instrument the variables correlated with
u i (x2 and z2 of eqs. 2 and 3). Deviations from the mean of x1 are used to produce
unbiased estimates for the time varying variables (x2) and the mean of x1 is used
as an instrument for the time-invariant variable (z2). While Hausman and Taylor
simply assume that x1 and z1 are uncorrelated with u, the applied researcher faces
the problem of distinguishing endogenous from exogenous right-hand side
variables. Moreover, it must not be the case that x1 and z1 are good instruments
for x2 and z2, respectively.
All three procedures used in the presence of time-invariant variables and unit
effects suffer from omitted variable bias (pooled OLS), are likely to be
inconsistent and biased (random effects), or have poor small sample properties
8
and leave the researchers with discrete choices that are hard to justify (HausmanTaylor).3
3. Fixed effects vector decomposition
In this section we suggest an alternative procedure for the estimation of timeinvariant variables in the presence of unit effects that maintains the small sample
properties and the unbiasedness of the fixed effects model. The logic of our
specification is straightforward: Unit fixed effects are a vector of the mean effect
of omitted variables, including the effect of time-invariant variables. Thus, in a
first stage we obtain the unit fixed effects vector by estimating a fe-model that
excludes the time-invariant variables. In a second stage, the vector can be
decomposed into a part explained by the time-invariant variables and an errorterm.4 In stage 3, this error-term accounts for the unobserved unit fixed effects
and, therefore, captures the potential of omitted variable bias. This enables us to
re-run stage 1 by pooled OLS. Since we include only one variable (the error term
of the second stage) to account for all remaining unobservable individual effects
in the third stage regression, we have to adjust the degrees of freedom.
In brief, the fixed effect vector decomposition technique carries out the following
three steps:
(1) estimation of the unit fixed effects by the baseline panel fixed effects
model excluding the time-invariant right hand side variables;
(2) regression of the fixed effects vector on the time invariant explanatory
variables of the original model (by OLS);
(3) estimation of a pooled OLS model by including all explanatory timevariant variables, the time-invariant variables and the unexplained part of
the fixed effects vector. This stage is required to control for
multicollinearity and to adjust the degrees of freedom.5
3
4
5
The poor quality of the instrument suggested has frequently raised some
concerns and alternative proposals. See Breusch, Mizon and Schmidt’s (1989)
proposal of feasible instruments and Amemiya and MaCurdy (1986).
Having said this, we hasten to admit that there is at least one potential source of
bias: As in all pure cross-sectional models, we cannot control for omitted variable
bias in the estimation of the time-invariant variables in the second stage. An
additional source of bias is the distribution of the fixed effects. If unit-fixed
effects are not normally distributed, OLS estimates may be biased if N is small .
Upon request we provide a STATA program (ado-file) that executes all three
steps and adjusts the variance-covariance matrix. Several options, like ar1 errorcorrection and robust VC-matrix are allowed.
9
This procedure adopts the robustness of the fixed effects model and allows for
correlation between explanatory variables and the unobserved individual effects.
The effects of time varying factors are consistently analyzed and remain unbiased.
Since fixed effects are robust with respect to potential correlation between right
hand side variables and the individual specific effects the exogeneity of
explanatory variables is not required.
In econometric terms, the fevd technique works as follows. Recall the datagenerating process of eqs. 1-3. The within estimator de-means the data and
removes the individual effects u i :
yit − yi = ( x it − x i ) ß + εit − εi ≡ yit = x itβ + εit
(7)
Thus, the fixed effects are
û i = yi − x iβˆ FE
(8)
In the second stage we regress the û i on the z-variables.
û i = ω + z i γ + ηi
(9)
where ϖ is the intercept of the stage 2 equation and ηi is the error. Note that we
get a biased estimate of γ if we exclude variables that are simultaneously
correlated with the unit-effects û i and the time-invariant variables z i . As one can
see, ηi is the part of u i that is not explained by the time invariant z-matrix.
In the third stage, we re-run the full model without the unit effects but including
the decomposed unit fixed effect vectors including ηi obtained in stage 2. This
stage is estimated by pooled OLS (or Prais-Winston in the presence of serial
correlation).
yit = α + x it β + z i γ + ηi + εit
(10)
ηi is no longer correlated with any of the z i , but by including the error term of
stage 2 we are able to account for individual specific effect that can not be
observed. The coefficient of ηi is either equal to 1.0 or at least close to 1.0 (by
accounting for serial correlation or panel heteroscedasticity) in stage 3.
Stage 3 is necessary for basically two reasons. On the one hand we must adjust the
degrees of freedom by u i -1 in calculating the variance-covariance matrix
of β and γ . Not correcting the degrees of freedom would lead to potentially severe
underestimation of standard errors and overconfidence in the results. In adjusting
10
the standard errors we explicitly control for the specific characteristics of the three
step approach.6 On the other hand, stage 3 also accounts for the potential
multicollinearity between the time-variant variables and the time-invariant
variables.
Estimating stage 3 by pooled OLS further requires that heteroscedasticity and
serial correlation must be eliminated beforehand. We suggest running a robust
Sandwich-estimator or/and model the dynamics by an MA1 process (PraisWinston transformation of the original data).7
At least in theory this method has three obvious advantages: a) the fixed effects
vector decomposition does not require prior knowledge of correlation between the
explanatory variables and the unit specific effects, b) the estimator relies on the
robustness of the within-transformation and does not need to meet the
orthogonality assumptions of random effects, and c) xtfevd maintains the
consistency and efficiency of OLS.
4. Monte Carlo Experiments
Given the properties of the unit effects decomposition technique there are good
reasons to believe that this estimator is superior to pooled OLS, random effects
and Hausman-Taylor if the empirical model includes time-invariant variables in
panel data. We now report a series of Monte Carlo experiments which aim at
assessing the performance of the available procedures in fairly small samples
which are typical for comparative politics. We will demonstrate that xtfevd does
better than its alternatives in situations common to applied political research.
Before discussing the results of the Monte Carlo experiments, however, we briefly
describe the design of the experiments.
Design of the Experiments
All experiments use simulated data, which are generated to discriminate between
the various estimators, while at the same time mimic some properties of panel
data. Specifically, the data generating process underlying our simulations is as
follows:
yit = α + β1x1it + β2 x2it + β3 x3it + β 4 z1i + β5 z2i + β6 z3i + u i + ε it
6
7
(11)
Upon request we provide a STATA program (ado-file) called xtfevd that executes
all three steps and adjusts the variance-covariance matrix.
The STATA ado allows for an ar1 error-correction and robust VC-matrix.
11
where the x-variables are time varying and the z-variables are time-invariant, both
groups are drawn from a normal distribution. Variables x3 and z3 are correlated
with the unit specific effect ui .8 Variables x1, x2, z1 and z2 are uncorrelated with
ui . We draw ui from a normal distribution in the first series of experiments and
from a gamma distribution in the second series of experiments. This allows to test
the sensitivity of the various procedures to unit effects that are not normally
distributed. In real data, unit fixed effects are unlikely to be normally distributed.
The idiosyncratic error εit is white noise and repeatedly drawn from a normal
distribution, and the R-squared is fixed at 50 percent for all experiments.9
Experiments are conducted with various combinations of N and T to reflect the
datasets that are typically analyzed in comparative politics. We use all
permutations of N=20, 30, 50, 80 and T=20, 40, 60, 100. For each combination of
T and N we conduct 1000 experiments. Finally, we hold the coefficients of the
true model constant throughout all experiments at the following values:
α = 1, β1 = 0.5, β2 = 2, β3 = −1.5, β4 = −0.4, β5 = 1.8, β6 = 4.5 .
Analysis of simulated data
We are interested in the bias and the efficiency of the estimators. We define bias
as the average deviation of the coefficients from the true coefficients,
()
K
(
b βˆ = ∑ βˆ − β true
k =1
)
K,
(12)
where k = {1, 2,..., K} is the number of simulations. For the purpose of this paper
we define efficiency as the standard deviation of the betas
( ) ( βˆ − β )
σ βˆ =
true
2
.
(13)
()
To begin, we consider the bias b βˆ of the first series of experiments where ui is
drawn from a normal distribution.
table 1 about here
8
9
We have experimented with various correlations, but since our results are robust
with respect to various correlation coefficients 0.5 ≤ ρ ( ui , x3i ; ui , z3i ) ≤ 0.95 , we do
not report the results from varying correlation coefficients here. All reported
results are therefore based on a correlation of app. 0.5.
For each permutation we generated 1000 replicates of εit and yit . Code was
written in STATA and is available upon request.
12
All procedures have difficulties in calculating the effect of time-invariant
variables that are correlated with the unit effects. The average deviations from the
true betas are smallest for the fixed effects vector decomposition technique and
largest for Hausman-Taylor. In fact, xtfevd is approximately 50 percent less biased
than the second best estimator, pooled OLS and random effects.
Hausman-Taylor is not only the most biased, but it also performs poorly with
respect to the time-invariant variables. It is the only specification that is likely to
report biased coefficients of time-invariant variables uncorrelated with the unit
effects. Since this result may come as a surprise it is important to note that we
correctly specified the endogenous variables. Bias surges if researchers make a
incorrect decision about exogeneity.
OLS and random effects models fail to give an unbiased coefficients of the timevariant variable that is correlated with the unit fixed effects (x3). This outcome is
expected because – having assumed unit effects correlated with x3 – we clearly
observe omitted variable bias here. Neither pooled OLS nor random effects
models capture this effect unless the number of observations approaches infinity
and random effects converge to the fixed effects estimator.
Figures 1.1-1.8 present the Kernel density distribution of the coefficients for x3
and z3 when N=30 and T=20. We have chosen this combination of N and T for
illustration purposes, since this represents a fairly common sample size in
comparative politics. The figures highlight the difficulties in obtaining the effect
of time-variant and time-invariant variables correlated with the unit effects.
Figure 1.1 -1.8 about here
Pooled OLS and random effects produce biased results for both x3 and z3. Both,
the fixed effects vector decomposition technique and Hausman-Taylor correctly
estimate the coefficient of the time-variant variable that co-varies with the unit
effects. Both models vary with respect to the estimation of z3, the time-invariant
variable that is correlated with the unit effects. The poor performance of
Hausman-Taylor in obtaining the coefficient of z3 stems from its small sample
inefficiency. Hausman-Taylor apparently requires very large samples to become
13
reliable in respect to time-invariant variables correlated with the unit effects.
Thus, while Hausman-Taylor and xtfevd are less biased than pooled OLS and
random effects, xtfevd has by far more favorable small sample properties than
Hausman-Taylor.
On average, the fixed effect vector decomposition technique behaves best. It is
almost as good as pooled OLS and random effects models in estimating the
coefficients of time-invariant variables and it provides less biased results than
both alternatives in calculating the effect of time-variant variables that are
uncorrelated with the unit fixed effects.
This finding is additionally supported by table 2 that reports the variance of the
estimates.
table 2 about here
Table 2 displays the standard deviation of the estimated betas from the true betas,
σ βˆ . Pooled OLS and random effects give a higher variance of betas than
()
Hausman-Taylor and xtfevd when calculating coefficients of time-invariant
variables. Since both methods also generate more biased coefficients for timeinvariant variables correlated with the unit effects, pooled OLS and random
effects should be avoided in this case. More favorable results occur for both
estimators when we consider time-invariant variables. This is clearly not the
domain of the Hausman-Taylor model, which has by far the largest variance of all
four procedures. In sum, xtfevd is least biased and most efficient – at least when
we analyze data that mirrors the data generating process underlying our Monte
Carlo experiments.
Sample size and bias
The preceding subsection has shown that on average and when analyzing wellbehaved data xtfevd performs better than its competitors. In this subsection we
examine whether the estimators’ bias and efficiency vary with the sample size.
We study the performance of the procedures for all permutations of
N = {20,30,50,80} and T = {20, 40, 60,100} . We focus on the results of the
‘problematic’ variables (x3 and z3) and do not further consider the variables
uncorrelated with the unit effects. In addition, we contrast the results of a model in
which unit effects are drawn from a normal distribution to the results of a model
14
where unit effects are drawn from a gamma distribution. Table 3 reports the bias
of estimators with respect to both the distribution of unit effects and the sample
size.
table 3 about here
The Hausman-Taylor and xtfevd procedures are superior in estimating the effects
of time-variant variables correlated with the unit effects. This holds true for all
combinations of N and T. Both OLS and random effects produce biased
coefficients when the time-variant variable is correlated with the unit fixed
effects. The estimates of xtfevd and Hausman-Taylor remain unbiased even if the
unit effects are gamma distributed. The impact of skewed unit effects on the bias
of coefficients is large for random effects and pooled OLS. On average, b βˆ of
()
pooled OLS and random effects almost triple when unit effects are skewed.
In our view, this constellation is likely to exist in real data. There is little reason to
believe that unit effects are normally distributed and it is likely that the unit
effects are correlated with the time-variant variables. Estimating the empirical
model by either pooled OLS or random effects cannot be recommended.
Pooled OLS and random effects work better when we consider time-invariant
variables correlated with the unit fixed effects. Table 4 shows that no procedure is
likely to produce unbiased coefficients for time-invariant variables correlated with
the unit effects, but Hausman-Taylor performs worse than the other three
alternatives.
table 4 about here
Bias decreases as the number of cross-sections increases but applied researchers
should nevertheless keep in mind that the coefficients of time-invariant variables
correlated with the unit effects are likely to be upward biased. In many cases this
knowledge will prevent applied researchers from making careless inferences.
The random effects specification works best for time-constant variables correlated
with individual effects and this holds especially true when the unit effects are
skewed. If the unit effects are drawn from a normal distribution the difference
15
between random effects models and xtfevd becomes small and lies within the
sampling variance. Most noteworthy, the Hausman-Taylor procedure is worse
than any alternative including pooled OLS. The bias is large regardless of the
distribution of unit effects. This outcome is probably due to a combination of poor
small sample properties and an insufficient quality of instruments. It appears that
using Hausman-Taylor has some similarity to organizing a coefficient lottery.
To sum up, all procedures give unbiased estimates of time-variant and timeinvariant variables that are uncorrelated with the unit-effects. In most real data
analyzed in comparative politics, the unit effects are likely to be correlated with
the explanatory variables. While xtfevd and Hausman-Taylor work much better
than random effects and pooled OLS when time-variant variables are correlated
with the unit effects, Hausman-Taylor is unreliable in cases where time-invariant
variables are correlated with unit effects. This possibility should not be dismissed.
Fortunately, xtfevd and random effects generate relatively unbiased coefficients
for time-invariant variables.
Under normal conditions, the fixed effects decomposition works in no respect
worse than any of its alternatives. Thus, we recommend that applied researchers
employ this procedure. Only if a) time-invariant variables are correlated with the
unit effects and b) the unit effects are skewed, researchers should avoid using
xtfevd.10 Under these circumstances the random effects model seems to be the
least biased choice. One should keep in mind, however, that random effects
obtains biased coefficients for time-variant variables correlated with unit-effects.
Sample size and efficiency
Small samples are prone to large variation in the estimated betas. This holds
especially true for the effect of time-invariant variables, because the number of
observation is T times smaller than the number of observations of time-variant
variables. Consequently, efficiency is a crucial issue.
The calculation of efficiency is based on the same experiments as the preceding
discussion of bias. Table 5 presents the estimators’ efficiency with respect to the
time-variant variable co-varying with the unit effects.
10
The output of xtfevd.ado provides information on the correlation between the unit
effects and the endogenous variables and on the skewness of the unit effects.
16
table 5 about here
Hausman-Taylor and xtfevd are more efficient in estimating the effect of timevariant variables correlated with the unit effects. The difference to random effects
and pooled OLS increases with the number of cross-sections. The larger N, the
more efficient Hausman-Taylor and xtfevd estimate time-variant variables. When
the unit effects are skewed, the efficiency of all procedures declines. The relative
advantage of Hausman-Taylor and xtfevd over random effects and pooled OLS
remains intact.
With the exception of Hausman-Taylor, efficiency is generally smaller for timeinvariant variables. The efficiency of Hausman-Taylor apparently depends on
large samples.
table 6 about here
As table 6 demonstrates, no procedure can adequately cope with the coincidence
of complications such as time-invariant variables correlated with skewed unit
effects. Applied researchers should keep this in mind when deriving inferences
from datasets where both unfortunate conditions are combined. Under any other
circumstance, researchers should find results produced by xtfevd acceptable. In the
absence of fully reliable solutions, conservatism in making inferences is
warranted.
5. Re-analyses
This section reports the results from replicating three contemporary analyses that
use pooled OLS in the presence of time-invariate variables: Evelyn Huber and
John D. Stephens study of the ‘Development and Crisis of the Welfare State’ and
Pablo Beramendi and Thomas Cusack (2004) analysis of the determinants of
income inequality. The aim of this section is to demonstrate that coefficients may,
but do not need to, vary if researchers apply more appropriate estimators for the
problem at hand.
Re-estimation of Huber/ Stephens 2001
Huber and Stephens aim at explaining the long-term patterns of welfare state
development in respect to the short- and long-term dynamics as well as the cross-
17
sectional variance.11 They take issue with the de facto Beck-Katz standard of
estimating panel data by OLS and including the lagged dependent variable (to
eliminate autocorrelation) plus unit fixed effects. They justify their deviation from
the standard procedure by claiming that the “analysis of short term change can
lead to quite misleading conclusions about long-term change.” (HS 2001: 4)
Specifically, analyses of short-term changes “are unable to explain the increasing
divergence of welfare states over time.” (HS 2001: 9)
Huber and Stephens argue that the long-term development of the welfare state has
been influenced by constitutionally created veto points. These veto points tend to
make welfare state expansion (but also welfare state retrenchment) more difficult
for the government. Political systems with more checks and balances should
maintain lower levels of government spending in general and also less social
security contributions in particular.
It is well understood among political scientists that the constitutional factors do
not vary much over time. Therefore, analyses of constitutional effects typically
have to deal with the problem of time-invariant variables in one way or another.
Huber and Stephens opt for aggregating constitutional features into a
‘constitutional structure’ variable that counts various veto points. Specifically,
their measure is a simple additive index including categorical data for federalism
(0,1,2), presidentialism (0,1), bicameralism (0,1,2) and the use of referenda (0,1).
This constitutionalism variable is ordinally scaled and ranks from 0 to 6. Since it
is almost time-invariant, they estimate the model by pooled OLS.
This research strategy has at least one obvious drawback: It is impossible to
identify the effect of each single constitutional variable. However, the impact of
the various constitutional features on government spending is likely to be unequal
across the various veto points. Implicitly, Huber and Stephens assume that the
effect of federalism and bicameralism on government spending and social security
transfers is exactly twice as large as the effects of presidentialism and referenda.
For that reason, we have not only re-analyzed the Huber-Stephens specification
(table 3.3. model 1) but also run a specification that uses the four original veto
points variables rather than the composite index.12
11
12
This part has profited from communication with Neal Beck.
We left Huber and Stephen’s estimation approach unchanged. This might be
problematic since the Im, Peseran, Shin test (2002) does not allow to reject the
hypothesis of a unit root. Keep in mind, however, that unit root test performs
18
Using STATA 8.0 and the data set that Stephens has posted on the Luxemburg
Income Study (LIS) webpage,13 we have not been able to exactly replicate the
results reported in the book by Huber and Stephens (p. 72-73). However, we
arrive at exactly the same numbers that John Stephens kindly provided in assisting
our replication attempts and – most importantly – these coefficients are Sala-iMartin-robust (signs of the coefficients do not differ) to the results reported by
Huber and Stephens.14 We re-analyze their specification based upon the updated
dataset and employing STATA instead of SHAZAM, which was used by Huber
and Stephens.15
Huber and Stephens regress government spending in 19 OECD countries on a
battery of political, social and economic variables. Their sample covers the years
1960 to 1994. However, for reasons beyond our knowledge, the model they report
analyzes only the years from 1960 to 1986. Table 7 presents the STATA output
that John Stephens kindly provided upon request and our re-analyzes using the
four procedures we found relevant in the presence of time-invariant variables.
table 7 about here
The coefficients reported by Huber and Stephens are fairly but not completely
robust. Our results deviate from those reported by Huber and Stephens in respect
to the effect of the population share of individuals older than 65 years, of
authoritarian legacy and of inflation. First, we find that the population share of
individuals above 65 has no significant effect on social security transfers. Second,
once we split Huber and Stephens’ composite constitutional constraints index into
its original components, we find that authoritarian legacy significantly reduces
social security transfers. Third, we also obtain a significant impact of inflation on
social security transfers if we control for unit effects. Finally, the composite index
13
14
15
poorly, and that failure to reject the unit root hypothesis does not imply that data
is non-stationary. We nevertheless add an appendix in which we estimate a model
in differences.
http://www.lisproject.org/publications/welfaredata/welfareaccess.htm.
As a matter of fact, the STATA results of the updated dataset exhibit more
significant relations between the exogenous and the endogenous variable and
provide larger support for Huber and Stephens theoretical claims than the models
reported in their book.
We kindly thank John Stephens for his support with the data and with the
replication analysis.
19
of Huber and Stephens conceals more than it reveals. The effect of constitutional
constraints on social security benefits is not homogenous. While federalism and
presidentialism reduces social security benefits as suggested by Huber and
Stephens theoretical claims, neither bicameralism nor referenda (a Switzerland
dummy) appear to be significantly related to social transfers.
Probably to the satisfaction of the authors, we find no difference to any of the
coefficients Huber and Stephens are theoretically interested in. Our analyses,
therefore, provide additional support for the theories more broadly discussed by
Huber and Stephens.16 However, we find significance for our assumption that the
aggregation of constitutional rules into a summary variable is not useful. To be
sure, Huber and Stephens correctly argue that veto points limit the observed rise
of the welfare state. But different constitutional settings do not have identical
effects. Bicameralism exerts the strongest influence on government spending,
followed by presidentialism. Neither federalism nor referenda are significantly
related to government spending if we rely upon fixed effect vector decomposition.
Re-estimation of Beramendi/ Cusack 2004
Pablo Beramendi and Thomas Cusack’s (unpublished) paper studies the
determinants of income inequality in OECD countries. Beramendi and Cusack
distinguish between three types of income, wages, total income from economic
activity and disposable income. Thus, their analyses allow isolating the
redistributive efforts of governments. We re-estimate model 11 of their discussion
paper version, which examines the determinants of disposable income, broadly
defined as market income plus government transfers minus taxes. The model controls for pre-tax inequality and also contains union density and ‘left government
inheritance’.
Model 11 includes one time-invariant variable, a dummy that controls for
coordinated market economies. The theoretical justification for the inclusion of
this variable is provided by Michael Wallerstein and Miriam Golden (2000) as
well as by Peter Hall and David Soskice (2001). In the view of these authors,
coordinated market economies are characterized by centralized wage negotiations
between unions and employer associations and possibly governments. Centralized
bargaining tends to moderate the unions’ demands for higher nominal wages. In
exchange for wage moderation the government provides social security systems to
16
See appendix A for a regressions in differences.
20
workers. As a consequence, coordinated market economies are associated with
more redistribution and less after tax and transfers income inequality.
Beramendi and Cusack justify running a pooled OLS model without eliminating
serial correlation of errors and controlling for unit effects in the following way:
“In the context of data sets in which the variance is dominated by the
‘between units’ (as opposed to the “within units”) component and where
there are time invariant independent variables, the inclusion of both FE and
a LDV frequently does more harm than good. Let us briefly explain why.
Consider first the inclusion of fixed effects in any specification with a time
invariant independent variable, such as the institutional terms in the wage
and disposable income inequality equations or the wage inequality measure
itself in the market income equation. In such cases, the specific value each
country takes in the independent variable is going to be highly collinear
with that country’s FE. As a result, the inclusion of FE yields substantively
uninteresting and generally misleading results in which none of the
variables of interest are statistically significant and yet the adjusted rsquared increases dramatically due to multicollinearity.” (Beramendi/
Cusack 2004: 29)
We do not agree, but that’s another issue (see Plümper et al. 2004).
Table 8 reports our re-analysis of the Beramendi-Cusack model. Note that we
simply do not have a sufficient number of time-invariant variables to be able to
estimate the Beramendi-Cusack model by the Hausman-Taylor formulation.
Hausman-Taylor requires at least as many exogenous time-invariant variables as
there are endogenous time-invariant variables.17
table 8 about here
In passing, we may say that – quite to our surprise – the information on the
disposable income inequality used by Beramendi and Cusack appears to be
stationary. It should be emphasized that the results reported by Beramendi and
Cusack are only mildly influenced by the switch from Panel OLS to both random
effects and fixed effects vector decomposition. While the former outcome is not
surprising given the similarity between Panel-OLS and random effects, the latter
result may come as a surprise. However, the time-variant explanatory variables
included in the Beramendi-Cusack model are uncorrelated with the unit effects.
The union density score shows the highest correlation with u i , but the correlation
17
Needless to say that this is another disadvantage of the Hausman-Taylor
procedure.
21
is only 0.37. All other variables are even less correlated with the unit fixed effects.
In any case, we find a smaller coefficient for the time-invariant variable – the
coordinated market economy dummy – but the interpretation of Beramendi and
Cusack remains intact. The negligible influence of controlling for unit fixed
effects may easily stem from the comparably low correlation between the unit
effects and the exogenous variables. As we know from OLS, excluded variables
cause little bias if they are orthogonal to the included exogenous variables. Their
results are also robust to the inclusion of additional or the exclusion of present
regressors.
Discussion of Re-Analyses
We draw two conclusions from our re-analyzes of two interesting and important
studies in comparative politics and comparative political economy. Controlling for
unit effects is essential in comparative politics. Political scientists are unlikely to
formulate theories and design empirical models that completely explain the
variance across nation-states. Even if we analyze a relatively homogenous group
of countries, for instance OECD countries, we cannot expect to capture all crosssectional variance by the exogenous variables. Neither should social scienstists
aim at formulating a complete model, we believe.
And this directly leads to our second conclusion: Large empirical models (as the
one suggested by Huber and Stephens) are likely to suffer from collinearity and
endogeneity. From our perspective, it is more promising to estimate smaller
empirical specifications that more closely represent the theoretical claims. In this
case, however, researchers almost must control for unit-effects and should run
estimators that do not require orthogonality between the unit effects and the
substantive explanatory variables. This is the domain of the unit effects vector
decomposition technique.
6. Conclusion
The results of the Monte Carlo experiments suggest that xtfevd is the least biased
estimator when time-variant and time-invariant variables are correlated with the
unit effects. This constellation appears to be common to social science data.
The unit fixed effects vector decomposition technique produces the least biased
and most efficient coefficients under a wide variety of data generating processes.
The main advantages of xtfevd come from its desirable small sample properties
and from its unbiasedness in estimating the coefficients of time-variant variables
that are correlated with the unit-effects. The efficiency of random effects models
22
and especially of the Hausman-Taylor procedure depend on large samples. The
three-stage vector decomposition technique shares the small sample properties
with pooled OLS without inheriting pooled OLS’s potential for bias.
In particular, when unit effects are uncorrelated with the time-variant variables all
four estimators are unbiased. When unit effects are uncorrelated with timeinvariant variables, pooled OLS, random effects and fixed effects vector
decomposition give unbiased, Hausman-Taylor biased estimates. When unit
effects are correlated with the time-variant variables, pooled OLS and random
effects model perform poorly, fixed effect vector decomposition and HausmanTaylor are unbiased, but Hausman-Taylor is less efficient. Finally, when unit
effects are correlated with time-invariant variables, all procedures are about
equally biased, but Hausman-Taylor is by far least efficient. In cases where both
time-variant and time-invariant variables are correlated with the unit effects, unit
fixed effects decomposition technique clearly outperforms its competitors.
However, one final clarification is in order: The vector decomposition technique
does slightly worse than the random effects model when the following conditions
are simultaneously given: a) the time-variant variables are uncorrelated with the
unit-effects, b) the time-invariant variables are correlated with the unit effects, and
c) the distribution of the unit effects is extremely skewed. In this case, the vector
decomposition technique suffers from biased estimates of the second stage.
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24
Appendix A: Huber-Stephens type of model in differences
lagged change in government
consumption
level variables
military spending
per capita income
federalism
presidentialism
referenda
authoritarian legacy
voter turnover
openness
fdi outflows
inflation
unemployment
old
female labor force participation
cumrelcb
left
left * female
difference variables
military spending
per capita income
voter turnover
openness
fdi outflows
strikes
inflation
unemployment
old
female labor force participation
cumrelcb
left
left * female
η
intercept
Im Pesaran Shin t-bar
F
R² within
F (stage2)
R² (stage 2)
corr ( u i , Xβ )
N
Estimated by xtfevd
coeff
0.0188
std. err.
0.0334
-0.0090
0.0000
0.0240
0.3970
0.0978
0.0730
0.0061
-0.0061
0.0305
-0.0106
-0.0165
-0.0092
-0.0233
0.0151
0.0135
-0.0006
0.0269
0.0000
0.0378
0.1030
0.1359
0.0597
0.0046
0.0020
0.0403
0.0213
0.0095
0.0182
0.0069
0.0069
0.0055
0.0004
0.5837
-0.0011
0.1976
-0.0063
-0.0429
0.0118
0.2271
-0.0161
0.8571
0.2663
0.0433
0.1194
0.0017
1.0000
0.7065
-3.830
27.25
0.6917
56.84
0.4216
-0.574
0.0642
0.0001
0.0838
0.0041
0.0528
0.0101
0.1192
0.0229
1.0175
0.1131
0.1023
0.0475
0.0058
0.2208
0.4412
384
****
***
*
***
**
**
****
****
**
*
**
**
****
25
pooled OLS
( )
b ( βˆ )
b ( βˆ )
b ( βˆ )
b ( βˆ )
b ( βˆ )
u normal distributed
xtfevd
xtreg, re
xthtaylor
b βˆ1
0.002
-0.005
0.013
-0.002
2
0.006
0.000
0.002
-0.002
3
0.295
-0.001
0.277
0.006
4
-0.014
-0.002
0.019
0.051
5
-0.016
-0.032
-0.015
0.166
6
0.335
0.415
0.356
0.550
0.668
0.455
0.682
0.777
sum b ( β k )
Table 1: Mean Deviation of estimated betas from true betas
( )
σ ( βˆ )
σ ( βˆ )
σ ( βˆ )
σ ( βˆ )
σ ( βˆ )
σ βˆ1
OLS
u normal distributed
xtreg, re
xtfevd
xthtaylor
0.276
0.279
0.280
0.279
2
0.083
0.082
0.083
0.082
3
0.324
0.132
0.320
0.135
4
0.217
0.206
0.230
0.427
5
0.255
0.328
0.289
0.524
6
0.388
0.474
0.400
1.300
Table 2: Efficiency of the estimated betas
26
OLS
N=20
N=30
N=50
N=80
xtfevd
N=20
N=30
N=50
N=80
xtre
N=20
N=30
N=50
N=80
xthtaylor
N=20
N=30
N=50
N=80
T=20
0.334
0.264
0.259
0.283
u normal distributed
T=40
T=60
0.244
0.358
0.309
0.289
0.240
0.269
0.391
0.316
T=100
0.191
0.204
0.336
0.429
T=20
0.883
0.790
0.989
0.985
u gamma distributed
T=40
T=60
0.722
0.653
1.052
0.879
0.967
0.874
1.012
0.958
T=100
1.065
0.860
0.943
0.855
-0.006
-0.005
0.003
-0.006
0.002
-0.006
-0.001
-0.001
-0.002
0.005
-0.005
-0.002
0.004
0.001
0.006
0.000
0.025
0.001
0.004
0.013
-0.022
-0.005
0.003
-0.004
0.000
-0.002
-0.002
-0.004
-0.009
-0.003
0.000
-0.005
0.170
0.297
0.475
0.330
0.253
0.135
0.308
0.312
0.180
0.144
0.337
0.205
0.176
0.399
0.372
0.339
0.703
0.897
0.732
1.009
0.902
0.949
0.932
0.894
0.477
0.896
0.842
0.921
0.493
0.918
0.798
0.903
0.012
0.014
0.006
0.005
-0.003
0.003
0.003
0.005
0.009
0.013
0.007
0.006
0.003
0.011
0.000
0.003
0.013
0.005
0.006
-0.003
0.056
-0.017
0.071
-0.001
0.017
0.008
0.005
0.008
-0.005
-0.002
0.000
-0.002
Table 3: Sample Size and Average Bias for x3 (time-variant, correlated with ui )
27
OLS
N=20
N=30
N=50
N=80
xtfevd
N=20
N=30
N=50
N=80
xtre
N=20
N=30
N=50
N=80
xthtaylor
N=20
N=30
N=50
N=80
T=20
0.456
0.233
0.338
0.381
u normal distributed
T=40
T=60
0.359
0.671
0.492
0.243
0.227
0.171
0.348
0.458
0.538
0.413
0.266
0.473
0.604
0.473
0.252
0.378
0.328
0.327
0.227
0.334
2.629
-0.036
0.320
-1.328
u gamma distributed
T=40
T=60
0.835
0.799
0.646
0.520
0.634
0.522
0.508
0.323
T=100
0.346
0.264
0.395
0.273
T=20
0.580
0.651
0.536
0.371
0.501
0.442
0.387
0.363
0.508
0.457
0.254
0.335
0.711
0.973
0.798
1.217
0.319
0.704
1.228
1.384
1.041
1.265
1.351
1.305
1.199
1.218
1.043
0.981
0.555
0.397
0.352
0.246
0.510
0.502
0.462
0.425
0.278
0.187
0.264
0.309
1.078
0.604
0.973
0.545
0.603
0.515
0.601
0.667
0.786
0.884
0.781
0.405
0.996
0.341
0.638
0.478
-0.722
0.671
0.679
1.499
0.403
0.538
0.656
1.489
0.480
0.434
0.927
0.170
-4.626
2.467
1.625
0.666
1.416
2.424
1.760
1.654
5.175
0.045
-0.599
0.121
2.254
-0.061
0.560
0.792
Table 4: Sample Size and Average Bias for z3 (time-invariant, correlated with ui )
T=100
0.362
0.527
0.561
0.681
28
OLS
N=20
N=30
N=50
N=80
xtfevd
N=20
N=30
N=50
N=80
xtre
N=20
N=30
N=50
N=80
xthtaylor
N=20
N=30
N=50
N=80
T=20
0.403
0.322
0.299
0.304
u normal distributed
T=40
T=60
0.290
0.379
0.333
0.305
0.257
0.279
0.397
0.321
0.242
0.213
0.154
0.131
0.171
0.141
0.111
0.083
0.297
0.361
0.497
0.349
0.263
0.206
0.157
0.120
u gamma distributed
T=40
T=60
0.779
0.697
1.070
0.896
0.979
0.883
1.017
0.962
T=100
0.215
0.218
0.340
0.430
T=20
0.963
0.855
1.014
1.000
0.140
0.109
0.087
0.063
0.104
0.086
0.062
0.039
0.524
0.409
0.331
0.257
0.355
0.282
0.214
0.174
0.292
0.237
0.179
0.140
0.220
0.179
0.145
0.103
0.297
0.187
0.324
0.322
0.228
0.181
0.348
0.213
0.203
0.410
0.378
0.342
0.824
0.943
0.775
1.024
0.944
0.972
0.946
0.903
0.536
0.913
0.854
0.926
0.532
0.930
0.806
0.907
0.176
0.150
0.106
0.081
0.145
0.117
0.078
0.064
0.107
0.084
0.061
0.043
0.539
0.402
0.318
0.261
0.358
0.308
0.233
0.178
0.305
0.278
0.190
0.141
0.226
0.175
0.140
0.102
Table 5: Sample Size and Efficiency for x3
T=100
1.075
0.868
0.947
0.858
29
OLS
N=20
N=30
N=50
N=80
xtfevd
N=20
N=30
N=50
N=80
xtre
N=20
N=30
N=50
N=80
xthtaylor
N=20
N=30
N=50
N=80
T=20
0.340
0.319
0.376
0.402
u normal distributed
T=40
T=60
0.385
0.687
0.510
0.276
0.256
0.207
0.365
0.467
0.614
0.651
0.317
0.491
0.630
0.502
0.278
0.393
0.410
0.383
0.273
0.366
3.013
0.798
1.016
2.434
u gamma distributed
T=40
T=60
0.909
0.873
0.729
0.556
0.672
0.557
0.537
0.354
T=100
0.370
0.289
0.405
0.286
T=20
0.733
0.798
0.608
0.431
0.530
0.464
0.402
0.377
0.528
0.468
0.272
0.444
0.852
1.077
0.859
1.246
0.486
0.745
1.256
1.398
1.079
1.294
1.364
1.315
1.238
1.233
1.053
0.988
0.595
0.419
0.378
0.269
0.538
0.518
0.473
0.437
0.306
0.222
0.280
0.320
1.235
0.712
1.042
0.590
0.717
0.592
0.640
0.688
0.827
0.924
0.805
0.429
1.021
0.385
0.658
0.494
0.868
0.770
0.778
1.781
0.542
0.606
0.701
1.546
0.586
0.626
0.950
0.280
15.006
2.594
1.967
0.896
3.455
3.505
2.114
2.232
5.777
0.619
0.912
1.165
2.302
0.606
0.824
1.141
Table 6: Sample Size and Efficiency for z3
T=100
0.437
0.552
0.581
0.691
30
depvar: ssbencor
procedure:
Left
Cumrelcb
Constr97
Results provided
by John Stephens
xtpcse
0.1699
(0.0655) ***
0.3785
(0.0629) ***
-1.1481
(0.2772) ***
xtfevd
0.1456
(0.0634) **
0.2743
(0.0656) ****
-0.9119
(0.2775) **
Fed
Pres
Strbic
Referen
Flfp
Left*fem
Vturn
Old
Strks
Authleg
Rgdpc
0.1125
(0.0441) ***
0.0181
(0.0047) ***
-0.0722
(0.0475)
0.6483
(0.1801) ***
-0.0799
(0.0229) ***
-0.6825
(0.5308)
0.0005
0.2382
(0.0590) ****
0.0214
(0.0046) ****
-0.0543
(0.0518)
0.0631
(0.2625)
-0.0433
(0.0210) **
-0.7689
(0.5665)
0.0004
xtfevd
0.1491
(0.0718) **
0.2995
(0.0817) ****
xtreg, re
0.1182
(0.0279) ***
0.4790
(0.0350) ***
xthtaylor
0.0615
(0.0534)
0.3397
(0.0502) ***
-1.2356
(0.6451) *
-3.8533
(1.6905) **
0.2268
(0.5469)
-2.4191
(2.2729)
0.2646
(0.0835) ***
0.0228
(0.0056) ****
-0.0829
(0.0632)
0.1878
(0.3056)
-0.0406
(0.0230) *
-1.8056
(0.9127) **
0.0003
-1.9215
(0.2162) ***
-1.1071
(0.5911) *
-0.8709
(0.2508) ***
-10.7307
(0.7432) ***
0.1647
(0.0332) ***
0.0277
(0.0025) ***
-0.1260
(0.0262) ***
0.5983
(0.1128) ***
-0.1790
(0.0102) ***
-1.9427
(0.3437) ***
0.0009
-0.8315
(1.5134)
-4.4428
(3.0768)
-3.1422
(0.7120) ***
-1.9580
(5.1658)
0.3095
(0.0536) ***
0.0253
(0.0033) ***
0.0150
(0.0540)
0.2885
(0.4792)
-0.1201
(0.0170) ***
-2.1188
(1.4868)
0.0009
31
Cpi
Unemp
Mil
Fdiout
Openness
η
Cons
rho
N
R²
Chi²
norm ( u i )
Jarque& Bera
norm ( ηi )
(0.0001) ***
0.2558
(0.2675)
0.4901
(0.0651) ***
0.1393
(0.1548)
-0.1806
(0.1451)
0.0028
(0.0140)
1.8956
(5.2914)
0.8564
416
0.530
806.28
(0.0001) ***
0.7350
(0.2360) ***
0.4597
(0.0585) ****
-0.0441
(0.1883)
-0.1179
(0.1522)
-0.0123
(0.0124)
0.9096
(0.1576) ****
0.5493
(5.8231)
0.8785
416
0.5017
(0.0001) **
0.7308
(0.2535) ***
0.4416
(0.0609) ****
-0.0729
(0.2133)
-0.1283
(0.1532)
-0.0099
(0.0130)
0.8542
(0.1918) ****
3.2458
(6.5881)
0.8941
416
0.4549
22.29 ****
22.29 ****
11.28 **
11.28 **
Jarque & Bera
F
27.12 ****
19.23 ****
Table 7: Replication Results for Huber/ Stephens 2001, p. 72-73, model 1.
(0.0001) ***
-0.5539
(0.1224) ***
0.5617
(0.0533) ***
-0.5496
(0.1554) ***
0.0582
(0.2128)
-0.0040
(0.0087)
(0.0001) ***
-0.1141
(0.1730)
0.6015
(0.0557) ***
-0.2448
(0.1780)
0.0671
(0.2173)
-0.0476
(0.0132) ***
11.1813
(2.7884) ***
0.0000
416
0.920
4545.44
-2.6110
(7.4181)
0.8806
416
3219.32
32
Market Inc.
Inequality
Union Density
Coordinated Mark.
Econ
Left Govern.
Inheritance
η
Intercept
rho
N
R²
F
norm ( u i )
Jarque& Bera
norm ( ηi )
Pooled OLS
(Beram. and
Cusack)
0.3265
(0.0694) ****
-0.0009
(0.0001) ****
-0.0420
(0.0053) ****
-0.0188
(0.0054) ***
Random Effects
0.2428
(0.0323) ****
0.2333
(0.0342) ****
41
0.901
78.67 ****
41
0.900
0.3465
(0.0725) ****
-0.0009
(0.0002) ****
-0.0407
(0.0085) ****
-0.0182
(0.0069) ***
fixed effect
vector
decomposition
0.3108
(0.0547) ****
-0.0016
(0.0002) ****
-0.0285
(0.0054) ****
-0.0205
(0.0050) ****
0.9967
(0.2361) ****
0.2751
(0.0266) ****
0.1183
41
0.917
89.50 ****
7.25 **
1.59
Jarque & Bera
Chi²
145.22 ****
Table 8: Replication Results for Beramendi/ Cusack 2004 (table 11)
33
Kernel density graphs for x3 (N30T20)
x3
z3
0
0
.5
.5
1
1
Density
1.5
Density
1.5
2
2
pooled OLS
-2
-1.5
b3
-1
-.5
-2
-1.5
b3
-1
-.5
-2
-1.5
b3
-1
-.5
-2
-1.5
b3
-1
-.5
4
4.5
b6
5
5.5
4
4.5
b6
5
5.5
2
Density
1.5
1
.5
0
0
.5
1
Density
1.5
2
xtfevd
-2.5
1
.5
0
0
.5
1
Density
1.5
Density
1.5
2
xtre
4
4.5
5
b6
5.5
6
6.5
0
0.1.2.3.4
.5
1
Density
1.5
Density
2
xthtaylor
-2.5
2
4
b6
6
8
10