This paper proposes a variant of the classical HAUSMAN specifi cation test commonly employed to decide whether the estimation of a random-eff ects model is a viable alternative to estimating fi xed eff ects. Whereas the classical test probes the equality of fi xed- and random eff ects, the proposed variant focuses on the equality of between-groups and fi xed-eff ects coeffi cients. While both test procedures lead to the same conclusions, the panel model specifi cation underlying our testing strategy facilitates the simultaneous estimation of the fi xed- and between-groups eff ects. As a consequence, we are able to examine both the equality of the whole range of coeffi cients as well as that of individual variables. The usefulness of the test is illustrated using a panel of household travel data for Germany.Specification tests, fuel price elasticity

Colin Vance

Manuel Frondel

English

Research Papers in Economics

Fixed, Random,
or Something in Between?
A Variant of HAUSMAN’s Speciﬁ  cation Test
for Panel Data Estimators
#160
RUHR
Manuel Frondel
Colin Vance
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ISSN 1864-4872 (online) – ISBN 978-3-86788-179-1
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Manuel Frondel and Colin Vance
Fixed, Random,
or Something in Between?
A Variant of HAUSMAN’s Speciﬁ  cation Test
for Panel Data EstimatorsRuhr Economic Papers #124
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ISSN 1864-4872 (online)
ISBN 978-3-86788-179-1Manuel Frondel and Colin Vance1
Fixed, Random, or Something in Between? –
A Variant of HAUSMAN’s Speciﬁ  cation Test for 
Panel Data Estimators
Abstract
This paper proposes a variant of the classical HAUSMAN speciﬁ   cation test com-
monly employed to decide whether the estimation of a random-eﬀ  ects model is a 
viable alternative to estimating ﬁ  xed eﬀ  ects. Whereas the classical test probes the 
equality of ﬁ  xed- and random eﬀ  ects, the proposed variant focuses on the equality 
of between-groups and ﬁ  xed-eﬀ  ects coeﬃ   cients. While both test procedures lead to 
the same conclusions, the panel model speciﬁ  cation underlying our testing strategy 
facilitates the simultaneous estimation of the ﬁ  xed- and between-groups eﬀ  ects. As 
a consequence, we are able to examine both the equality of the whole range of coef-
ﬁ  cients as well as that of individual variables. The usefulness of the test is illustrated 
using a panel of household travel data for Germany.
JEL Classiﬁ  cation: C12
Keywords: Speciﬁ  cation tests, fuel price elasticity
January 2010
1   M a n u e l  F r o n d e l ,  R W I ,  R u h r - U n i v e r s i t ä t  B o c h u m ;   C o l i n  V a n c e ,  R W I ,  J a c o b s  U n i v e r s i t y  
Bremen. – We are grateful for invaluable comments and suggestions by Walter Krämer and 
Christoph M. Schmidt. – All correspondence to Manuel Frondel, RWI, Hohenzollernstr. 1-3, 
45128 Essen, Germany, e-mail: frondel@rwi-essen.de.1 Introduction
The econometric modeling of panel data typically applies two principal approaches,
ﬁxed- and random-effects estimators. In the ﬁxed-effects approach, time-invariant, un-
observable factors for each observation unit are either explicitly captured by dummy
variables or wiped out through time-demeaning. In contrast, these time-invariant un-
observables are treated as part of the disturbances in the random-effects model, the-
reby assuming that their correlation with the regressors is zero. If this assumption is
met, the random-effects estimator confers the advantage of greater efﬁciency over the
ﬁxed-effects estimator. Violation of the assumption, however, implies biased estimates.
Toinvestigatetheappropriatenessofeitherofthesetwoapproaches, HAUSMAN’s
(1978) speciﬁcation test is commonly employed. It is based on the idea that the set of
coefﬁcient estimates obtained from the ﬁxed-effects estimation – taken as a group –
should not differ systematically from the set derived via random-effects estimation
under the null hypothesis that the unobservable, individual-speciﬁc effects and the
regressors are orthogonal. If the test results suggest rejecting the equality of both co-
efﬁcient sets, applied researchers generally proceed to draw conclusions based on the
ﬁxed-effects estimates. This course of action effectively results in the wholesale discar-
ding of the random-effects estimates. In such cases, however, it would be frequently
interesting to know whether the inequality holds for the complete set of coefﬁcients or
whether there are exemptions for speciﬁc variables of interest.
To examine the equivalence of the coefﬁcients for individual variables, this pa-
per suggests a variant of the HAUSMAN test that is based on the fact that testing the
equality of ﬁxed- and random effects is numerically identical to testing the equality
of between-groups and ﬁxed effects (HAUSMAN and TAYLOR, 1981). Speciﬁcally, we
show that using a straightforward model speciﬁcation allows us to simultaneously
estimate the ﬁxed- and between-groups effects – either on the basis of Ordinary (OLS)
or Generalized Least Squares (GLS) – and to test both the equality of coefﬁcients for
individual variables as well as that of the whole range of coefﬁcients.
4The following section presents the theoretical basis of the test variant. Its useful-
ness is illustrated in Section 3 using a panel of household travel diary data for Germa-
ny. The last section summarizes and concludes.
2 Methodology
Mostappliedpanelanalyseseschewthebetween-groupsestimator,asitfailstocapture
inter-temporal information, and instead focus on ﬁxed- and random effects. Yet, using
the fact that testing the equality of ﬁxed- and random effects is numerically equivalent
to testing that the set of ﬁxed effects, w, equals the set of between-groups effects, b, the
between-groups estimator plays a major role in our approach.
Proposition 1: Departing from a standard panel data model,
yit = β0 + β
Txit + ξi + νit,i =1 ,...,N,t =1 ,...,T, (1)
where ξi denotes an unknown individual-speciﬁc term and νit is a random-error com-
ponent that varies over individuals i and time t, and estimating the speciﬁcation
yit = β0 + w
T(xit − ¯ xi)+b
T¯ xi + ξi + νit, (2)
via OLS simultaneously yields estimates of the between-groups and ﬁxed effects, whe-
re the OLS estimator of w provides for the ﬁxed-effects estimates and the estimates of
the between-groups effects are given by the OLS estimator of b.
Proof:First,thebetween-groupsestimatorofparametervectorβ emergingfrommodel
(1) can be obtained by averaging (1) over time and estimating the result via OLS:
¯ yi = β0 + β
T¯ xi + ξi +¯ νi, (3)
where ¯ yi, ¯ xi and ¯ νi denote the time means of yit,xit and νit, respectively. Second, the
ﬁxed-effects estimates of β can be retrieved by subtracting (3) from (1) and estimating
the result via OLS:
yit − ¯ yi = β
T(xit − ¯ xi)+νit − ¯ νi. (4)
5Third, instead of estimating either (3) or (4), we alternatively suggest in Proposi-
tion 1 estimating (2) via OLS to at once get both the ﬁxed- and between-groups effects.
This can be seen as follows: Upon averaging (2) over time, the term related to w was-
hes out so that the result is, aside from the notation of the parameter vector, identical
to (3):
¯ yi = β0 + w
T · 0 + b
T¯ xi + ξi +¯ νi = β0 + b
T¯ xi + ξi +¯ νi. (5)
Therefore, averaging either (1) or (2) over time and estimating the results via OLS must
yield the same estimates, namely those of the between-groups effects.
Finally, subtracting (5) from (2) yields (4), with w instead of β as the parameter
vector:
yit − ¯ yi = w(xit − ¯ xi)+νit − ¯ νi. (6)
In other words, either demeaning (1) or (2) and estimating the result via OLS provides
for the ﬁxed-effects estimates.
Proposition 2: Estimating speciﬁcation (2) via GLS yields exactly the same results as
the OLS estimation of this speciﬁcation.
Proof: First, estimating panel model (1) via GLS is equivalent to estimating
yit − λ · ¯ yi = β0 · (1 − λ)+β
T · (xit − λ · ¯ xi)+ξi − λ · ξi + νit − λ · ¯ νi (7)
via OLS (see e.g. WOOLDRIDGE 2008:490), where λ is a parameter that is determined
by the time horizon and the variances of the error terms ξi and νit. Second, employing
the same transformation to the modiﬁed speciﬁcation (2) yields:
yit−λ·¯ yi = β0·(1−λ)+w
T(xit−¯ xi)−λ·w
T·(xit − ¯ xi)+b
T·¯ xi−λ·b
T·xi+ξi−λ·ξi+νit−λ·¯ νi.
(8)
Recognizing that (xit − ¯ xi)=0and xi = ¯ xi and rearranging gives:
yit = β0 + w
T(xit − ¯ xi)+b
T¯ xi + ξi + νit + λ · (¯ yi − β0 − b
T¯ xi − ξi − ¯ νi)
  
=0
,
where the last bracket vanishes because of equation (3). In short, both transformation
(8) and speciﬁcation (2) are identical and, hence, the OLS estimation of speciﬁcation
(2) delivers the same results as estimating transformation (8) via OLS, which is in turn
equivalent to estimating speciﬁcation (2) via GLS.
63 Empirical Example
To demonstrate the usefulness of the test, we employ household data drawn from the
German Mobility Panel (MOP 2010) and estimate fuel price elasticities using the follo-
wing speciﬁcation suggested by Proposition 1:
ln(eit)=β +bp ·ln(pi)+wp ·(ln(pit)−ln(pi))+bx
T · ¯ xi +wx
T ·(xit − ¯ xi)+ξi +νit , (9)
where ln(e) is the logged monthly fuel consumption, ln(p) denotes logged real fuel
price per liter and x designates a vector of control variables such as age of the car and
whether it is a premium make. A detailed data description can be found in FRONDEL,
PETERS, and VANCE (2008) or FRONDEL and VANCE (2009).
The advantage of estimating speciﬁcation (9), irrespective of whether OLS or GLS
is used, is that it allows us to at once retrieve the entire set of between-groups and ﬁxed
effects and, hence, to easily examine both the equality of the coefﬁcients for individual
variables on the basis of ordinary t-tests, as well as the equality of the whole range of
coefﬁcients using an F-test1:
H0 : bp = wp,bx = wx, (10)
where wp and wx designate the ﬁxed effects and bp and bx the between-groups effects,
respectively. According to HAUSMAN and TAYLOR (1981), any rejection of the null hy-
pothesis H0 also implies that the ﬁxed- and random effects are different and, hence,
that the ﬁxed effects should be preferred over the random effects.
In our empirical example, the result of the standard HAUSMAN test reported in
Table 1 indicates that the orthogonality hypothesis of the unobservable individual-
speciﬁc effects and the regressors is rejected. From the t-test results reported in the last
column, it becomes obvious that the reason for the rejection of the null is primarily
due to the difference in the estimates of just two variables: the number of employed
1KRÄMER and SONNBERGER (1986:98-99) show for the classical regression model y = Xβ + Zγ,
where γ =0under H0 and γ  =0under the alternative, that the F-test is identical to the HAUSMAN test
if the number of variables included in Z is equal to or smaller than those included in X.
7household members and the indicator for whether a vacation was taken during the
survey period.
Table 1: OLS Estimates of Speciﬁcation (9) and Test Results.2
Fixed Effects Between-Groups Effects t-Test
ln(e) Coeff.s Std. Errors Coeff.s Std. Errors Statistics
ln(p) ∗ -0.569 (0.166) ∗∗-0.605 (0.188) -0.21
car age -0.010 (0.007) ∗ -0.015 (0.006) -0.78
household size -0.002 (0.048) 0.016 (0.028) 0.38
children -0.020 (0.118) 0.045 (0.079) 0.66
# high school diploma 0.002 (0.055) 0.085 (0.036) 1.59
# employed ∗∗ -0.117 (0.044) ∗∗ 0.217 (0.037) ∗∗ 7.76
job change ∗∗ 0.116 (0.070) 0.115 (0.077) 0.00
vacation with car ∗∗ 0.254 (0.037) ∗∗ 0.435 (0.068) ∗∗ 3.55
diesel car -0.297 (0.196) 0.026 (0.089) 1.09
premium car 0.149 (0.149) ∗∗ 0.326 (0.052) 1.90
Standard HAUSMAN Test: χ2 (10) = ∗∗ 57.34
Note: ∗ denotes signiﬁcance at the 5 %-level and ∗∗ at the 1 %-level, respectively. Number of
observations (households) used for the estimations: 1,341 (530).
In contrast, the between-groups and ﬁxed-effects estimates of the key variable,
ln(p), do not signiﬁcantly differ from each other. This also suggests that the respective
ﬁxed- and random-effects estimates are equal in statistical terms, a suggestion that is
substantiated by the closeness of the ﬁxed-effects estimate of -0.569 and the (unrepor-
ted) random-effects estimate of -0.579. Furthermore, it can be empirically demonstra-
ted that using STATA’s ﬁxed- and between-groups effects estimation options precisely
2To correct for the non-independence of repeated observations from the same households over the
years of the survey, the regression disturbance terms are clustered at the household level. The presented
measures of statistical signiﬁcance are robust to this survey design feature. The t-test statistics are calcu-
lated using t =( bxk − wxk)/

 Va r(bxk)+ Va r(wxk), where the covariance Cov(bxk,w xk) vanishes due
to the orthogonality of the vectors ¯ xi and (xit − ¯ xi) pertaining to bx and wx, respectively.
8reproduces the OLS estimates displayed in Table 1, as is claimed in Proposition 1.3 It
also bears noting that these OLS estimates are identical to those obtained when using
STATA’s random-effects estimation option, which is in line with Proposition 2’s claim
that both the OLS and GLS estimates of speciﬁcation (2) are equal.
4 Summary and Conclusion
The HAUSMAN (1978) speciﬁcation test is commonly employed for selecting between
the ﬁxed- and random-effects estimators for panel data. The random-effects estimator
is based on the assumption that the correlation between the regressors and the unob-
servable, individual-speciﬁc effects is zero, a situation that should be considered the
exception rather than the rule (WOOLDRIDGE 2008:493). It is therefore not surprising
that this null hypothesis is frequently not found to withstand empirical scrutiny. If the
test statistic, which contrasts the ﬁxed- and random-effects estimates, rejects the null,
applied researchers generally discard the random effects and base their conclusions on
the ﬁxed-effects estimates.
This all-or-nothing choice prompted HAUSMAN and TAYLOR (1981) to propose
a model that introduces an instrumental variable estimator using both between- and
within-groups variation to correct for the correlation of selected regressors with the
individual effect. Using a straightforward model speciﬁcation that also draws on the
between- and within-groups variation, we suggest a test variant that is based on the
contrast of between-groups and ﬁxed effects and allows us to examine both the equa-
lity of the whole sets of coefﬁcients as well as that of individual variables, an issue that
cannot be addressed on the basis of the standard HAUSMAN test.
3The employed data set and code is available from the authors upon request. It should be noted
that STATA’s between-groups estimates and the OLS estimates of b of speciﬁcation (2) are not perfectly
identical if the employed panel is unbalanced, as in our example. In this case, one has to use weighted
least squares (WLS) in order to correct for the frequency of a household’s occurrence in the panel.
9References
FRONDEL,M .,V ANCE, C. (2009) Do High Oil Prices Matter? Evidence on the Mobility
Behavior of German Households. Environmental and Resource Economics, 43 (1), 81-94.
FRONDEL,M .,P ETERS,J .,V ANCE, C. (2008) Identifying the Rebound: Evidence from
a German Household Panel. The Energy Journal, 29 (4), 154-163.
KRÄMER,W .,S ONNBERGER, H. (1986) The Linear Regression Model under Test. Physica,
Heidelberg, Wien.
MOP (2010) German Mobility Panel. http://www.ifv.uni-karlsruhe.de/MOP.html
HAUSMAN, J. A. (1978) Speciﬁcation Tests in Econometrics Econometrica 46, 1251-1271.
HAUSMAN,J .A .,T AYLOR, W. E. (1981) Panel Data and Unobservable Individual Ef-
fects Econometrica 49, 1377-1398.
WOOLDRIDGE, J. M. (2008) Introductory Econometrics. Fourth Edition, South Western.
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Fixed, Random, or Something in Between? – A Variant of HAUSMAN’s Specifi cation Test for Panel Data Estimators

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