Negative values for estimated variances can arise in a panel data context. Empirical and theoretical literature dismisses the problem as not serious and a practical solution is to replace negative variances by its boundary value, i.e. zero. While this is not a concern when the individual variance components is "small" with respect to idiosyncratic variance component (making it indistinguishable from zero in practice), we claim that a negative estimated variance can also arise with a "large" individual variance component, when the orthogonality condition between the individual effects and regressors fails. Estimation problems are considered in the (feasible) generalized least squares and maximum likelihood frameworks.Panel data, random effect estimation, negative variances, maximum likelihood

Giorgio Calzolari

Laura Magazzini

English

Research Papers in Economics

Working Paper Series
Department of Economics
University of Verona
Negative variance estimates in panel data models
Laura Magazzini, Giorgio Calzolari
WP Number: 15 October 2010
ISSN:    2036-2919 (paper),    2036-4679 (online)Negative variance estimates in panel data models
Laura Magazzini∗ Giorgio Calzolari†‡
This version: October 26, 2010
Abstract
Negative values for estimated variances can arise in a panel data context.
Empirical and theoretical literature dismisses the problem as not serious and
a practical solution is to replace negative variances by its boundary value,
i.e. zero. While this is not a concern when the individual variance compo-
nents is “small” with respect to idiosyncratic variance component (making it
indistinguishable from zero in practice), we claim that a negative estimated
variance can also arise with a “large” individual variance component, when
the orthogonality condition between the individual eﬀects and regressors fails.
Estimation problems are considered in the (feasible) generalized least squares
and maximum likelihood frameworks.
Keywords: Panel data, random eﬀect estimation, negative variances, maxi-
mum likelihood.
∗Department of Economics, Universit` a di Verona, Viale dell’Universit` a 4, 37129 Verona, Italy,
<laura.magazzini@univr.it>
†Department of Statistics, Universit` a di Firenze, Viale Morgagni 59, 50134 Firenze, Italy,
<calzolar@ds.uniﬁ.it>
‡We gratefully acknowledge suggestions from D. Lubian, A. Sembenelli, but retain full respon-
sibility for the contents of this paper.
11 Background
The possibility of estimating a negative variance in the random eﬀect (RE) model
for panel data is an established problem in the empirical and theoretical litera-
ture. Econometrics textbooks acknowledge this possibility and refer to the work of
Maddala (1971), who considers maximum likelihood (ML) estimation in the case
of normally distributed error components and sets forth the condition under which
the estimated variance of the individual component is negative.1 If non-negativity
constraints are imposed for estimation, a boundary solution at zero can therefore
arise. Analogously, when estimation is performed by two-step feasible generalized
least squares (FGLS), the estimated variance of the individual component is not
guaranteed to be non-negative.2
Maddala and Mount (1973) explores the performance of the ML and alternative
FGLS estimators of the variance components by means of Monte Carlo simulations,
showing that the problem of negative variances exists within a range of methods
proposed in the literature. However, in their setting, an estimated negative variance
arises when the individual eﬀect variance is “small” relative to the variance of the
idiosyncratic component (of the order of 1/400). In this case, OLS estimation (that
sets to zero the variance of the individual component) provides results that are “just
as reliable” (Maddala and Mount, 1973).3 Accordingly, negative variance estimates
are deemed as “not serious” in the literature, and the problem is solved by replacing
1Maddala (1971) also shows that the estimate of the variance of the idiosyncratic component
cannot be lower than zero.
2Note that, under normality, (F)GLS and ML estimators are asymptotically equivalent. See
e.g. Hsiao (2003, §3.3); Greene (2003, §13.4); Cameron and Trivedi (2005, §21.7); Baltagi (2008,
§2.3).
3 Under the assumptions of the RE model, the choice between FGLS and OLS estimation is
only a matter of eﬃciency, where the FGLS estimator allows to fully exploit the information about
the covariance structure of the error terms.
2non-positive estimates with the boundary value, i.e. zero.4
Interest in the issue has waned over time, even though the explanation provided
by the literature is incomplete (and sometimes potentially misleading).
In this paper, we show that a negative value for the estimated variance of the
individual component (both within the FGLS and ML framework) can arise if the
orthogonality condition between the individual eﬀects and the regressors is not sat-
isﬁed. The assumption is required for the consistency of the FGLS (and ML) es-
timators, and the problem is related to the well-known property of the estimated
variance of a model with endogenous regressors: if endogeneity is not accounted for,
the estimated variance of the error term is downward biased. Indeed, the estimated
variance of the individual component is obtained as the diﬀerence between (i) the
average of the squared OLS residuals (or the residuals from the between regression,
both aﬀected by the endogeneity problem in case the orthogonality condition is not
satisﬁed), and (ii) the average squared residuals from the within regression (robust
to any pattern of correlation between the regressors and the individual eﬀect). As
a result, if the orthogonality condition is not satisﬁed, the estimated value of the
individual variance will be smaller than the true value, possibly also lower than zero.
2 Why a negative estimated variance?
The RE panel data model estimated on N units (ﬁrms, individuals, households,...)
over T time periods (i = 1     N; t = 1     T) can be written as:
y = Xβ + u (1)
4Maddala and Mount (1973); Cameron and Trivedi (2005, §21.7); Hsiao (2003, §3.3); Baltagi
(2008, §2.3).
3where y is a NT ×1 vector, X is a NT ×k matrix on k observable strictly exogenous
variables, β is a k × 1 vector that contains the parameters of interest, and u is an
NT × 1 vector of disturbances.
We decompose the disturbance terms into two independent components:5
uit =  i + νit (2)
where  i represents the individual eﬀects assumed to be IN(0 σ2
µ), and vit are
assumed IN(0 σ2
ν).
Note that σ2
µ is not uniquely interpreted as the variance of the individual eﬀect
(bounded to be greater than 0), but it can also be interpreted as the covariance
between uit and uis (t  = s).
2.1 Maddala (1971)’s condition
Maddala (1971) considers the following reparametrization: σ2 = σ2
µ + σ2
ν, and ρ =
σ2
µ σ2.
In his seminal paper, the likelihood equations are solved by a two-step procedure
and a necessary and suﬃcient condition is identiﬁed for the occurrence of a boundary
solution at ρ = 0.6 This is:
Tyy − α
′Txxα > T[Byy − 2α
′Bxy + α
′Bxxα] (3)
5More generally time eﬀects can be also considered, leading to uit =  i + τt + νit with the
additional assumption that τt, the time eﬀects, are distributed as IN(0 σ2
τ). As standard panel
dimensions allow the inclusion of time dummies, time eﬀect are omitted from the error term (i.e.
σ2
τ = 0).
6Note that the reformulation of the model is only valid if the covariance between the two error
components is equal to zero, as it is customarily assumed in panel data applications. Berzeg
(1979) relaxes this assumption letting cov( i νit) = σµν. Accordingly the sum of the two variance
components is σ2 = σ2
µ + σ2
ν + 2σµν, and ρ = (σ2
µ + 2σµν) σ2, which admits negative values if
σ2
µ < −2σµν.
4where α = T −1
xx Txy, Tyy =
PN
i=1 y′
iyi, Txx =
PN
i=1 X′
iXi, Bxx = 1
T
PN
i=1(X′
iee′Xi),
Byy = 1
T
PN
i=1(y′
iee′yi), and Bxy = 1
T
PN
i=1(X′
iee′yi), with Xi is a T × k matrix
containing observations on unit i, yi is the T × 1 vector of observations for the
dependent variable (unit i), and e is a T × 1 vector of ones.
In its original form (3), Maddala’s condition is simply an algebraic inequality
involving OLS estimated coeﬃcients and transformation of the data. With some
algebra, condition (3) reduces to:7
N X
i=1
T X
t=1
X
s =t
ˆ uisˆ uit < 0 (4)
where ˆ uit denotes OLS residuals.
The formulation that uses OLS residuals (4) shows us that Maddala (1971)’s con-
dition in equation (3) does not provide any hints on the reasons behind a (meaning-
less) negative estimated variance. Equation (3) and (4) simply state “tautologically”
that the ML estimate of the individual variance is negative if it can be estimated
as a negative value. Indeed, the within group correlation of estimated residuals is
exploited in order to build FGLS/ML estimates of the individual variance compo-
nent.8
7Computations exploit the orthogonality between OLS estimated residuals and the explanatory
variables included in the model. Detailed algebra is available from the authors upon request.
8When discussing the FGLS estimation of the random components model, Wooldridge (2002,
§10.4) exploits the (degree of freedom adjusted average) within-group correlation of pooled OLS
residuals (4) in order to get an estimate of σ2
µ. Indeed, under the assumption of uncorrelated
random components, E[
PT
t=1
P
s =t uituis] = σ2
µT(T − 1). Based on this expression, Wooldridge
(2002, §10.4) claims that a negative ˆ σ2
µ can be explained in the case of negative serial correlation in
uit. If this is the case, E
hPT
t=1
P
s =t uituis
i
= σ2
µT(T − 1) + σ2
ν
PT
t=1
P
s =t ρts, with ρts denoting
the correlation between νit and νis. For the expected value to be negative, we therefore need:
σ2
µ σ2
ν < −
P
s =t ρts T(T −1) (coherently a negative variance can only arise in the case of negative
serial correlation). As an example, let us consider T = 5 and let νit follow an autoregressive process
of order 1 with ρ = −0 9, i.e. ρts = − 9|t−s|. In order to have an expected negative value, we
would need σ2
µ σ2
ν < 0 2, i.e. also in this case σ2
µ needs to be “small” with respect to σ2
ν.
52.2 Failure of the orthogonality condition
Starting from model (1) with uit =  i + νit, alternative ways of estimating the
variance components have been proposed in the literature, exploiting diﬀerent sums
of squared residuals (Wallace and Hussain, 1969; Amemiya, 1971; Maddala, 1971;
Swamy and Arora, 1972; Fuller and Battese, 1974). As an example, the estimate
of σ2
µ can be recovered by taking the diﬀerence between (a) the residuals from OLS
regression whose average sum of squares is exploited to estimate σ2
µ + σ2
ν, and (b)
the residuals from the within transformed regression (LSDV model) whose average
sum of squares is exploited to estimate σ2
ν (Greene, 2003, §13.4). The estimate of
σ2
µ is therefore expected to be non-negative as “the sum of squares in the LSDV
model cannot be larger than that in the simple regression with only one constant
term” (Greene, 2003, §13.4). Despite that, negative estimated variances can arise
in applications.9
Odd at a ﬁrst glance, the fact can be reconciled as OLS needs an additional
restriction with respect to LSDV for consistency, i.e. the orthogonality condition.
If this is not satisﬁed, the OLS (restricted) model suﬀers of the standard omitted
variable bias (due to the omission of the unit speciﬁc components), and the estimate
of the sum of the variance components is downward biased.10 On the contrary, the
estimated variance of the idiosyncratic component will rely on the residuals from
the within regression, that is not aﬀected by the presence of correlated individual
eﬀects. As a result, the estimate of the individual variance component ˆ σ2
µ will be
downward biased, possibly also lower than zero.11
9When non-negativity is imposed, a boundary solution at zero arises.
10Bounded to lie above zero as computed as the sum of squared OLS residuals.
11As uit =  i+νit, also the presence of correlation between ν and x would cause a downward bias
in the estimate of the sum of the two variance components that rely on OLS residuals. However,
this eﬀect would also bias the estimated variance of the within regression (b), making a negative
value less likely to appear.
6Within a ML framework, computations analogous to the FGLS formulas are
employed, without the degree of freedom correction for the estimated variance com-
ponents (Greene, 2003, §13.4). By relying on standard textbook treatment, the ML
estimate of σ2
µ is obtained by considering the diﬀerence between the variance of be-
tween residuals d σ2
ν T + σ2
µ and the variance of within residuals ˆ σ2
ν T. It is possible
to show that the diﬀerence is negative if and only if
N X
i=1
T X
t=1
X
s =t
ˆ ˆ uitˆ ˆ uis < 0 (5)
with ˆ ˆ uit denoting the ML (FGLS) residuals.12 Still, the between-regression provides
a consistent estimate of β if the orthogonality condition is satisﬁed, otherwise the
model suﬀers of an omitted variable bias and the estimated sum of squared resid-
uals is downward biased. Again, the estimate of the variance of within residuals
ˆ σ2
ν T is not aﬀected by the presence of correlation between x and  i, and therefore
the diﬀerence between the two, that is the estimated value of the variance of the
individual component, will be downward biased, maybe also lower than zero.
3 A Simulation Experiment
In this section we focus on the correlation structure between x and the individual
eﬀect  , and explore how this can aﬀect the estimated value of σ2
µ.
The model is simulated as follows (i = 1     N; t = 1     T):
yit = xit +  i + νit
12See e.g. Arellano (2003, §3.2) and Greene (2003, §13.4). Note the analogy with equation (4)
obtained by developing Maddala’s condition (3). Recall that OLS and FGLS only diﬀer in terms
of eﬃciency.
7Value of γ
0 0.1 0.25 0.5 0.75 0.95 0.99
Static DGP for x
Mean of ˆ σ2
bet 1.199 1.151 .9483 .5733 .3333 .2203 .2034
Mean of ˆ σ2
ν .9996 .9996 .9996 .9996 .9996 .9996 .9996
% of negative ˆ σ2
µ 0% 0% 0% 0% 0% 2.6% 37.8%
Dynamic DGP for x
Mean of ˆ σ2
bet 1.198 1.134 .8886 .5076 .3034 .2159 .2031
Mean of ˆ σ2
ν 1.001 1.001 1.001 1.001 1.001 1.001 1.001
% of negative ˆ σ2
µ 0% 0% 0% 0% 0% 7.1% 39.5%
Table 1: Results of Monte Carlo experiments
with  i ∼ N(0 σ2
µ), νit ∼ N(0 σ2
ν), and, in order to allow for correlation between x
and  , we let xit = γ i +
√
1 − γ2ξit with ξit ∼ N(0 1) independent of  i. As the
variables in a panel dataset may exhibit strong patterns of autocorrelation over time,
we also consider a dynamic speciﬁcation for x where we let xit = γ i+
√
1 − γ2ξit+
0 5xit−1.13 We let σ2
µ = σ2
ν = 1.
The data generating process is simulated for diﬀerent values of γ, selected in the
interval [0 1).14 We considered N = 1 000 and T = 5.
Standard formulas (Baltagi, 2008, pag. 19) are applied to compute ˆ σ2
µ and
ˆ σ2
ν. The sum of squared residuals of the between regression is used to estimate
σ2
bet = σ2
ν T +σ2
µ, and the sum of squared residuals from LSDV estimation provides
an estimate of σ2
ν. The estimated value of σ2
µ is therefore obtained by considering
ˆ σ2
bet − ˆ σ2
ν T.
Results of the Monte Carlo experiments are reported in Table 1 as a function of
the value of γ employed during the simulations.
For γ > 0, the estimated value of the variance from the between regression is
13In the dynamic speciﬁcation, we let xi0 = γ i (1 − 0 5) + ξi0. We simulate 10 + T values for
xit and then disregard the ﬁrst 10.
1410,000 Monte Carlo replications are considered. The seed is reset after each simulation set, so
that the diﬀerences among the distributions of the estimates are only driven by diﬀerences in γ.
8increasingly biased toward zero (bias is increasing with γ), whereas the estimated
variance of the idiosyncratic component (based on the FE estimator of β) is not
aﬀected by the increased correlation. As a result, the variance of the individual
component, estimated as the diﬀerence between estimated variance of the between
regression and the estimated variance of the idiosyncratic component is biased to-
ward zero, and can also assume negative values. When γ = 0 95, in 263 cases out of
10,000 Monte Carlo replications with static x the variance of the individual compo-
nents is estimated to be lower than zero (717 cases in the case of autocorrelated x)
and the number increases to 3782 when γ = 0 99 (3956 in the dynamic speciﬁcation
for x).
3.1 Further extensions
This problem has also implications for the computation of the Hausman statistics
that is used to discriminate between the RE and FE approaches (Hausman, 1978).
In applications, the estimated variance covariance matrix involved in the computa-
tion of the Hausman statistic can be not positive deﬁnite, and in more extreme cases
the value of the statistic is negative! Textbook explanation for the negative result
relies on a small sample problem, as the distribution of the Hausman statistic is
chi-squared asymptotically. We claim that it is possible to observe a (meaningless)
negative value for the Hausman statistics (or an estimated variance covariance ma-
trix that is not positive deﬁnite) when the orthogonality condition is not satisﬁed.
As shown, the correlation between the individual eﬀect and the regressors causes
a downward biased estimate of the sum of the two individual error components,
that can, in turn, lead to a non-positive deﬁnite estimate of the variance covariance
matrix of the diﬀerence of the FE and RE estimate of β. Recent research shows
9that the Hausman test can be negative even asymptotically if the alternative hy-
pothesis is correct, i.e., in our context, if the orthogonality conditions is not satisﬁed
(Schreiber, 2008). Schreiber (2008) also shows that in some cases, the pitfall can
lead to misleading positive test statistics.
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11

A comparative study of alternative estimators for variance components models used in econometric applications,

Analysis of Panel Data,

Econometric Analysis of Cross Section and Panel Data,

Econometric analysis of panel data,

Econometric analysis, 6th ed.,

Estimation of Linear Models with CrossedError Structure,

Microeconometrics,

Panel Data Econometrics,

The Error Component Model. Conditions for the Existence of the Maximum Likelihood Estimates,

The Estimation of the Variances in a Variance-Components Model,

The exact ﬁnite sample properties of the estimators of coeﬃcients in the error components regression models,

The Hausman Test Statistic can be Negative even Asymptotically,

The Use of Error Components Models in Combining Cross Section with Time Series Data,

Negative variance estimates in panel data models

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Negative variance estimates in panel data models

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