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Estimating individual effects and their spatial
spillovers in linear panel data models: Public capital
spillovers after all?
Karen Miranda
Óscar Martínez-Ibáñez
Miguel Manjón-Antolín

Document de treball n.26- 2015

DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

UNIVERSITAT
ROVIRA I VIRGILI
DEPARTAMENT D’ECONOMIA 

Edita:
Departament d’Economia
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Email: sde@urv.cat

CREIP
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Universitat Rovira i Virgili
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Adreçar comentaris al Departament d’Economia / CREIP

Dipòsit Legal: T – 1414 - 2015
ISSN edició en paper: 1576 - 3382
ISSN edició electrònica: 1988 - 0820

DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

Estimating individual eﬀects and their spatial spillovers in
linear panel data models:
Public capital spillovers after all?∗
, K. Miranda, O. Mart´
ınez-Iba˜ez, and M. Manj´n-Antol´
n
o
ın
QURE-CREIP Department of Economics, Rovira i Virgili University

Abstract
Individual-speciﬁc eﬀects and their spatial spillovers are not generally identiﬁed in linear
panel data models. In this paper we present identiﬁcation conditions under the assumption
that covariates are correlated with the individual-speciﬁc eﬀects and derive appropriate
GLS and IV estimators for the resulting correlated random eﬀects spatial panel data model.
We also illustrate the proposed estimators using a Cobb-Douglas production function
speciﬁcation and US state-level data from Munnell (1990). As in previous studies, we
ﬁnd no evidence of public capital spillovers. However, public capital does play a role in
the positive “outwards” spatial contagion of the individual eﬀects.

Keywords: correlated random eﬀects, spatial spillovers, panel data
JEL Classiﬁcation: C23, R11
∗

We thank P. Elhorst, A. P´rez-Laborda, Jesus Mur and participants at the ERSA Summer School 2016,
e
18 INFER annual conference, IAAE 2016 annual conference, 4th WIPE workshop, 40th Simposio de An´lisis
a
Econ´mico, and 8th Jean Paelinck seminar on Spatial Econometrics for helpful comments. This research was
o
funded by grants ECO2014-55553-P (Ministerio de Econom´ y Competitividad) and 2014FI B00301 (Agaur,
ıa
Generalitat de Catalunya). Usual caveats apply.
th

1

Introduction

Does public capital have an eﬀect on private output? And if it does, does this eﬀect spill over
nearby geographical areas? Using US state (and/or county) panel data, production function
estimates have consistently concluded that public capital and its spatially weighted counterpart
are not statistically signiﬁcant.1 In contrast, studies using alternative methodologies (e.g.,
VAR models), seem to suggest otherwise (Pereira and Andraz, 2013). In this paper we provide
production function estimates supporting the existence of public capital spillovers. To be
precise, we ﬁnd no evidence of a direct positive eﬀect of public capital on private output.
However, we ﬁnd evidence of a relation between public capital and the unobserved productivity
of the states (i.e., the individual speciﬁc eﬀect of the production function) and its spatial
spillover.2
To obtain these results, this paper introduces a correlated-random eﬀects model (Mundlak,
1978; Chamberlain, 1982) that presents spatial correlation in the individual eﬀects. To our
knowledge, only the random eﬀects model of Kapoor et al. (2007) accounts for this spatial
correlation. In the ﬁxed eﬀects case, Beer and Riedl (2012) advocate using an extension of
the Spatial Durbin Model for panel data that includes the spatially weighted individual eﬀects.
Ultimately, however, they argue that “it is (...) advisable to remove the spatial lag of the ﬁxed
eﬀects from the equation as the inclusion of both, [the individual eﬀects] and [their spatial
lags], leads to perfect multicollinearity” (p. 302). Removing the spatial lag of the ﬁxed eﬀects
does not generally preclude the consistent estimation of the parameters of the model (see e.g.
Halleck Vega and Elhorst, 2015). However, this practice rules out obtaining an estimate of the
individual-speciﬁc eﬀects (net of the spatially weighted eﬀects).3
1

See e.g. Munnell (1990); Baltagi and Pinnoi (1995); Holtz-Eakin and Schwartz (1995); Garcia-Mila et al.
(1996); and Kelejian and Robinson (1997).
2

As Boarnet (1998, p. 381-382) points out, “[p]ublic capital is provided at a particular place, and if such
capital is productive, it enhances the comparative advantage of that location relative to other places”. Also,
“productive public capital might shift economic activity from one location to another”.
3

This is a critical issue, for example, in two-step models that use this estimate as the dependent variable
(Combes and Gobillon, 2015). Similarly, obtaining an estimate of the spatial spillovers of the individual-speciﬁc
eﬀects may be of great interest (e.g., for assessing their geographical distribution, which is what we do in our
empirical application).

1

This raises the question of whether both individual eﬀects and their spatial spillovers can
indeed be identiﬁed in linear panel data models. In this paper we provide identifying conditions
in a model speciﬁcation that spatially weights both the independent variables and the individual
eﬀects. In particular, we show that there is no identiﬁcation problem if the covariates are
correlated with the individual-speciﬁc eﬀects and the individual eﬀects correspond to deviations
from the constant term.
Having proved that the model is identiﬁed, we then consider the estimation of its parameters
under alternative exogeneity assumptions on the explanatory variables. Under the assumption
that all the explanatory variables are strictly exogenous (with respect to the idiosyncratic
term), we derive a Feasible Generalised Least Squares (FGLS) estimator. We also prove that,
regardless of the structure of the variance-covariance matrix of the correlation functions shocks,
this estimator coincides with the within (ﬁxed eﬀects) estimator when all the explanatory
variables are used to construct the correlation functions. Under the assumption that the
explanatory variables are predetermined, we propose an Instrumental Variables (IV) estimator
to address the endogeneity of the means of the predetermined explanatory variables used
to approximate the correlation functions. We also advocate using the backward means of
these variables (i.e., the means taken, for each period, over only current and past values) as
instruments.
Lastly, we use these estimators and a (correlated random eﬀects) production function
speciﬁcation to address the existence of capital spillovers. Using the data and (a spatially
weighted variant of) the speciﬁcation used by Munnell (1990), we ﬁnd that, under strict
exogeneity, our FGLS estimates of a Cobb-Douglas production function for the US states over
the period 1970 to 1986 are largely consistent with those reported in related studies (using this
data set, as e.g. Baltagi and Pinnoi 1995; Kelejian and Robinson 1997; and using analogous
data sets, as e.g. Holtz-Eakin and Schwartz 1995; Garcia-Mila et al. 1996).4 However, when
we explore the possibility that (some of) the explanatory variables are not exogenous, we ﬁnd
evidence of predeterminedness in the public capital. We then estimate the model by IV to
4

The data set we employ is publicly available and can be downloaded, for example, from the Ecdat package
in R (a standardised binary contiguity spatial weights matrix of the US states is also included in the package).

2

ﬁnd that, under a sequential exogeneity assumption, states with a larger/smaller estimated
individual eﬀect tend to have larger/smaller negative spatial spillovers. In particular, we
consider both “spill-in” and “spill-out” eﬀects (LeSage and Chih, 2016), although only the spillout eﬀects turn out to be statistically signiﬁcant. Also, while the part of the individual eﬀects
associated with the private capital produces negative spatial contagion, the part associated
with the public capital produces positive spatial contagion. Consistent with previous literature,
however, we ﬁnd no signiﬁcant spatial spillovers in the public capital.
The rest of the paper is organised as follows. In Section 2 we discuss the identiﬁcation
problem and show that, under mild rank conditions, the correlated random eﬀects model
considered is identiﬁed. In Section 3 we present appropriate (FGLS and IV) estimators. In
Section 4 we present empirical evidence based on the work of Munnell (1990). Section 5
concludes.

2
2.1

Speciﬁcation and identiﬁcation of the model
Spatial spillovers and the identiﬁcation problem

Let us consider the spatial−(X, Ψ) panel data model, that is, the spatial (lag of) X model for
panel data with spatially weighted ﬁxed eﬀects:
y = Xβ + W Xγ + Ψµ + W Ψα + ε,

(2.1)

where y = (y11 , . . . , y1T , . . . , yN 1 , . . . , yN T ) is the dependent variable (as usual, i = 1, . . . , N
denotes cross-sectional, geographical units and t = 1, . . . , T denotes the time dimension),
X = (x11 , . . . , x1T , x21 , . . . , x2T , . . . , xN 1 , . . . , xN T ) is the N T × K matrix of explanatory
variables (i.e., xit is a row vector of order K) is the N T × K matrix of explanatory variables,
2
and ε is a zero-mean idiosyncratic error term with assumed variance-covariance matrix σε I N T ,

3

with I N T being the N T × N T identity matrix.5
We assume that neighbourhood relations do not change over time, so the spatial matrix is
W = w ⊗ I T , with I T denoting the T × T identity matrix and w = [wij ] being the N × N
spatial weight matrix that describes the spatial arrangement of the units in the sample. Also,
unobservable individual-speciﬁc eﬀects are collected in Ψµ, with Ψ = I N ⊗ ιT , I N being the
N × N identity matrix and ιT a vector of ones of order T . Notice that, in contrast to the
so-called SLX model (Halleck Vega and Elhorst, 2015), this model speciﬁcation accounts for
the spatial weights of the individual eﬀects through the term W Ψα. Thus, the parameters
2
of the model are β, γ, µ, α and σε . This means that 2(K + N ) + 1 parameters need to be

estimated.
The main motivation behind the use of this model speciﬁcation is the estimation of the
individual eﬀects and their spatial spillovers, since these often have a meaningful interpretation
(Combes and Gobillon, 2015). In particular, following LeSage and Pace (2009), we deﬁne
the spatial spillovers of the individual-speciﬁc eﬀects in terms of the partial derivative of the
(conditional expectation of the) dependent variable,
∂E [y | X, Ψ]
= (I N µj + wαj ) ⊗ I T
∂Ψj

(2.2)

where Ψj is the j-th column of Ψ and j = 1, . . . , N .
The oﬀ-diagonal elements of this matrix of partial derivatives represent the spillovers or
indirect eﬀects of unit j, whereas the diagonal elements of the matrix represent the direct
eﬀects of unit j. Notice, however, that since these eﬀects are time-invariant, we can, without
loss of generality, concentrate on the term in brackets, I N µj +wαj . Thus, a generic oﬀ-diagonal
element of this matrix, wil αj with i = l, measures the eﬀect of unit l having the unobservable
characteristics of unit j on the dependent variable of unit i. Similarly, a generic element of
the diagonal of this matrix, µj + wii αj , measures the eﬀect of unit i having the unobservable
characteristics of unit j on the dependent variable of unit i. This direct eﬀect reduces to µj for
5

Throughout the paper, we assume that a balanced (complete) panel data is available. However, results can
easily be extended to incomplete panels.

4

all i when wii = 0, the standard case.6
The vector of parameters µ has thus a neat interpretation. However, the role the spatial
weight matrix w and the vector of parameters α play in the spill-over eﬀects deserves some
further attention. On the one hand, the structure of w provides the deﬁnition of neighbourhood.
That is, which units are aﬀected (the “spill-out” eﬀects) and which are aﬀecting (the “spill-in”
eﬀects) by the spillover and, in distance-based matrices, how much will be aﬀected/aﬀecting
each unit. Thus, diﬀerent spatial weight matrices yield diﬀerent spatial spillovers. On the other
hand, the parameter αj provides a measure of the spatial contagion of the individual eﬀect of
unit j irrespective of the number of neighbours it has and how close/distant they are. That
is, αj provides a measure of the “potentiality of the spatial contagion” associated with the
individual eﬀect of unit j. We thus refer to α as the “potential” of the spatial spillovers of the
individual eﬀects.
We illustrate the calculation and interpretation of these direct and spillover eﬀects in the
empirical application of section 4. In any case, what is clear is that the estimation of these
eﬀects requires that of the parameter vectors µ and α (since w is assumed to be a ﬁxed and/or
known matrix). It is therefore critical to determine whether these (and the rest of) parameters
of the model are identiﬁed. Proposition 1 shows that, in general, this is not the case.
Proposition 1. The Spatial (Lag of ) −(X, Ψ) model for panel data with spatially weighted
ﬁxed eﬀects is not identiﬁed for any spatial weight matrix w.
Proof. See appendix.
Notice that, as Beer and Riedl (2012) argue, the omission of W Ψα does not preclude the
consistent estimation of the parameters of the model. Thus, if the spatial spillovers of the
individual-speciﬁc eﬀects are of no interest for the application in hand, their suggestion to
6

However, LeSage and Pace (2009) do not recommend reporting unit-level eﬀects but scalar summary
measures. Namely, the average of the main diagonal elements (direct eﬀects) and the cumulative sum of
the oﬀ-diagonal elements from each row, averaged over all rows (indirect eﬀects). In particular, if the spatial
weight matrix w is row-standarised and has zeros in the diagonal, these scalar summary measures correspond
to µ and α, respectively. We discuss the use of alternative unit-level indirect eﬀects (analogous to the ones
proposed by LeSage and Chih 2016) in section 4.

5

remove one of the components – i.e., either Ψ or W Ψ – is perfectly sensible. This is because
the model
y = Xβ + W Xγ + Ψµ∗ + ε,

(2.3)

with Ψµ∗ = Ψµ + W Ψα, is observationally equivalent to (2.1).
On the other hand, if the individual-speciﬁc eﬀects and/or their spatial spillovers are of
some interest, then the identiﬁcation and estimation of the model need to be discussed. To this
end, we go on to propose using a spatial−(X, Ψ) panel data model with correlated random
eﬀects. In particular, we show that, under mild assumptions, the model is identiﬁed. Later we
present appropriate estimators under alternative exogeneity assumptions

2.2

X
The Correlated Random Eﬀects Spatial−(X , Ψ ) Panel Data
Model

Fixed eﬀects models implicitly assume that the individual eﬀects are correlated with the
covariates. But they somehow ignore this correlation in the estimation procedure. In fact,
what the within and analogous transformations do (see e.g. Beer and Riedl, 2012) is to wipe
out the individual eﬀects so that this correlation is no longer a concern for the consistent
estimation of the model. An alternative procedure for obtaining consistent estimates, however,
is to incorporate this correlation into the model (Mundlak, 1978; Chamberlain, 1982). This is
the approach followed here. In particular, we make use of the correlation between covariates
and the (spatially weighted) individual eﬀects to identify the spatial contagion in the individual
eﬀects.
Our modelling approach is related to that of Debarsy (2012), who uses a correlated random
eﬀects speciﬁcation to construct an LR test on “the relevance of the random eﬀects approach”
(p. 112).7 Notice, however, that although we both deal with the correlation between individual
eﬀects and covariates, our purposes diﬀer markedly: while he seeks to correctly specify this
correlation, we use it as a means to identify the spatial contagion in the individual eﬀects. We
7

Although this is not the aim of this paper, an analogous Wald test could be developed using our model
speciﬁcation and estimation procedures.

6

also diﬀer in the model speciﬁcation which, although similar, treats the spatial contagion of
the individual eﬀects diﬀerently. Debarsy (2012) assumes that the individual eﬀects depend
on both the explanatory variables and the explanatory variables in their neighbourhood, but
there is no spatial contagion in the individual eﬀects. In contrast, we account for the spatial
contagion in the individual eﬀects (i.e., both the individual eﬀects and their spatial spillovers
are included in the speciﬁcation) and assume that the individual eﬀects and the “potential” of
their spatial spillovers depend on the (mean of the) explanatory variables, which allows us to
identify both the individual eﬀects and their spatial spillovers. These alternative assumptions
yield diﬀerent error component structures: a one-way error in his case, a two-way error in ours
(the additional component being a spatially weighted element).8
Thus, we assume the following relation between µ, α and the explanatory variables:
1
Ψ X ∗ Πµ + υ µ
T
1
α = Ψ XΠα + υ α ,
T
µ=

(2.4)

where Πµ and Πα are (K +1)×1 and K ×1 parameter vectors to be estimated, respectively, and
X∗ =

ιN T

X

contains the covariates and a vector of ones. Notice that a constant term

needs to be included in one of the equations in (2.4) to guarantee identiﬁcation, since the spatial
contagion of any common factor in the individual eﬀects µ (in particular, a constant term) is
not identiﬁed. Ultimately, this means that we are implicitly assuming that the individual eﬀects
correspond to deviations from the constant term. Also, the error terms υ µ and υ α are assumed
2
2
to be random vectors of dimension N with υ µ ∼ (0, σµ I N ) and υ α ∼ (0, σα I N ). However, υ µ

and υ α are not assumed to be independent, the covariance parameter, σµα , being such that
E(υ µ υ α ) = σµα I N with E denoting the mathematical expectation.
8

Another important diﬀerence with the work of Debarsy (2012) is that whereas he analyses the Spatial
Durbin Model (as Beer and Riedl 2012 do), our results are derived for the spatial −(X, Ψ) model. This allows
us to address the identiﬁcation and estimation of the model in a linear setting, whereas considering correlated
random eﬀects in a Spatial Durbin speciﬁcation would result in a non-linear model in which identiﬁcation
and estimation are more involved. (Debarsy (2012, p. 115), for example, “assume(s) that all parameters are
identiﬁed”; see, however, Lee and Yu 2016).

7

Plugging equations in (2.4) into the model (2.1) we obtain
y = Xβ + W Xγ +

1
1
ΨΨ X ∗ Πµ + W ΨΨ XΠα + η,
T
T

(2.5)

where η = Ψυ µ + W Ψυ α + ε. Notice that the resulting error component is similar to
the one proposed by Kapoor et al. (2007) in that both error components allow for spatial
contagion in the individual (random) eﬀects. It is diﬀerent in that while we assume that the
idiosyncratic term is not spatially correlated (and propose an identiﬁcation strategy that takes
into consideration that the individual eﬀects may have spatial eﬀects), Kapoor et al. (2007)
assume that the idiosyncratic term is spatially autocorrelated.
It is also interesting to note that our model speciﬁcation does not impose the existence
of spatial contagion in the individual eﬀects. In fact, there is no contagion at all if both Πα
and σα are zero (while there would still be some “random contagion” if Πα is zero but σα is
not). Similarly, the model does not impose correlation between the individual eﬀects and the
covariates. In fact, the speciﬁcation becomes that of a pure random eﬀects model if both Πµ and
Πα are zero (still with spatial contagion if σα is not zero). Thus, it is the statistical signiﬁcance
of these (sets of) parameters what ultimately determines the existence of spatial contagion in
the individual eﬀects and correlation between the individual eﬀects and the covariates.
Finally, in contrast to the spatial −(X, Ψ) model for panel data in (2.1), Proposition 2 shows
that the correlated random eﬀects spatial panel data model in (2.5) is generally identiﬁed.
Proposition 2. The correlated random eﬀects spatial panel data model in (2.5) is identiﬁed if
the matrix X =

X WX

1
ΨΨ
T

X∗

1
W ΨΨ
T

X

has full column rank.

Proof. See appendix.
Notice that since the number of parameters in the model is 4K + 1 (excluding the σ s),
N T ≥ 4K + 1 is a necessary identiﬁcation condition. Notice also that since ΨΨ X ∗ is a
N T × K + 1 matrix and W ΨΨ X is an N T × K matrix, both with row rank equal to N ,
N ≥ 2K + 1 is an additional necessary identiﬁcation condition. Further, X, W X, Ψ X ∗
and W Ψ X must have full rank. Lastly, time-invariant regressors must be included in either

8

the vector of explanatory variables (X and/or W X) or in the vector of determinants of the
individual eﬀects and their spatial spillovers (ΨΨ X ∗ and/or W ΨΨ X). Otherwise, there is
exact multicollinearity between the explanatory variables.

3

Estimation

We start by noticing that consistent estimation of the parameters of the model does not depend
on what the structure of the error term η is. In particular, assuming that the covariates are
strictly exogenous (meaning here that E(εit |xj1 , xj2 , . . . , xjT ) = 0 for all i, j), Ordinary Least
Squares (OLS) estimates of (2.5) are consistent.9 Yet a more eﬃcient Generalized Least Squares
(GLS) estimator can be derived by accounting for the error components structure of the model.
To be precise, the GLS estimator does not provide eﬃciency gains in the β and γ parameters
of the correlated random eﬀects model in (2.5). In fact, as shown below (see also Mundlak 1978),
the GLS estimator of these parameters coincides with the OLS estimator in (2.5) and the within
or ﬁxed-eﬀects estimator of the (observationally equivalent) model in (2.3). This means that,
if these parameters are the only ones of interest, eﬃciency considerations do not justify the
use of the GLS estimator. In contrast, the GLS estimator of (2.5) may provide eﬃciency gains
(with respect to OLS) in the Πµ and Πα parameters of the correlation functions. This is why,
under the above strict exogeneity assumption, we propose using a FGLS estimator based on
the estimates of the parameters of the variance-covariance matrix of η.
On the other hand, none of these estimators are consistent if the strict exogeneity assumption
does not hold. In particular, the presence of predetermined variables among the regressors (X
and W X) makes such variables endogenous when they are included among the variables that
compose the correlation functions in (2.4). To obtain consistent estimates, we propose an IV
estimator and the means of the endogenous variables taken, for each period, over only current
and past values (backward means) as instruments.
9

Notice that the spatial structure of our model requires an orthogonality condition involving not only all
the time periods (a standard assumption in applied work; see e.g. Wooldridge 2002) but also all the units.
1
Otherwise, we cannot guarantee the exogeneity of W X and T W ΨΨ X.

9

Next we discuss the derivation of the proposed estimators in detail.

3.1

GLS estimation under strict exogeneity

Given our initial assumption of spherical disturbances and the stochastic assumptions about
the behaviour of υ µ and υ α , the error-component η = Ψυ µ + W Ψυ α + ε has zero mean and
variance-covariance matrix given by
2
2
2
Ω = σµ ΨΨ + σα W ΨΨ W + σµα ΨΨ W + σµα W ΨΨ + σε I N T
2
2
2
= Ψ σµ I N + σα ww + σµα w + σµα w Ψ + σε I N T

(3.1)

Knowledge of this matrix suﬃces to derive the GLS estimator (see e.g. Wooldridge 2002) of
the parameters of the correlated random eﬀects model in (2.5). In particular, Mundlak (1978)
proves that, when all the explanatory variables are used to construct the correlation functions
and η = Ψυ µ + ε, this coincides with the within (ﬁxed eﬀects) estimator of β and γ in (2.3).
Next we generalise this result to any (non-spherical) variance-covariance matrix of υ µ .
Proposition 3. Consider the following correlated random eﬀects model:
y = Xλ +

1
ΨΨ X∗ Π + η
T

with X∗ being the matrix X plus a column of ones, λ and Π vectors of parameters with the
2
appropriate dimension, and E (ηη ) = Ω = ΨΣυ Ψ + σε I N T , where Συ is any variance-

covariance matrix. The GLS, OLS and within (ﬁxed eﬀects) estimators of λ are the same.

Proof. See appendix.
Notice that our model corresponds to X =

X WX

1
, T ΨΨ X =

1
ΨΨ
T

X

1
W ΨΨ
T

X

2
2
λ = (β, γ) and Συ = σµ I N + σα ww + σµα w + σµα w . Thus, Proposition 3 fully applies. More

generally, the previous proof shows that, regardless of the structure of the variance-covariance

10

,

matrix of υ µ and υ α , the GLS (and OLS) estimator of β and γ coincides with the within (ﬁxed
eﬀects) estimator.
Next we consider the derivation of the feasible version of this GLS estimator. This basically
2
2
2
requires a consistent estimate of the vector of parameters σ = (σµ , σα , σµα , σε ). To this end,

we notice that each component of Ω can be written as a linear function of σ:
E [ηit ηls ] = σM ilts ,

(3.2)

where i, l = 1, . . . , N and t, s = 1, . . . , T . Also, E [ηit ηls ] denotes the mathematical expectation
of ηit ηls and M is a 4 × 1 vector whose rows are functions of w. More speciﬁcally,
N
2
2
2
E ηit = σµ + σα

2
2
wij + 2σµα wii + σε

(3.3)

2
wij + 2σµα wii for t = s

(3.4)

j=1
N

E [ηit ηis ] =

2
σµ

+

2
σα
j=1

N
2
E [ηit ηls ] = σα

wij wlj + σµα (wil + wli ) for i = l

(3.5)

j=1

This allows us to consider the following linear regression to estimate σ:
ηit ηls = σM ilts + uilts
ˆ ˆ

(3.6)

ˆ
where η is obtained as the residual term of a consistent estimation of the model in (2.5). Given
the assumption of strict exogeneity of the covariates, OLS may be used for this purpose.
Under mild conditions, OLS estimation of (3.6) provides consistent estimates of σ (denoted
ˆ
by σ) and, with these in hand, we can obtain the FGLS estimates of the model.10

In

ˆ
particular, σ allows us to obtain ΩGLS using (3.1) and, by Cholesky Decomposition of its
10

2
2
2
Alternatively, one may impose the positiveness of the variances (σµ , σα and that σε ) and that the correlation
2
2
between µ and α lies in the [−1, 1] interval (−1 ≤ σµα × (σµ × σα )−1/2 ≤ 1), and use e.g a Non-Linear Least
Squares estimator of σ. Notice that this diﬀers from the approach followed by e.g. Kapoor et al. (2007) in that
their estimating equations are non-linear in the parameters of interest and they therefore have to resort to a
Generalized Moment estimator (which can also be used here).

11

−1

inverse, ΩGLS = D GLS D GLS , the transformation matrix D GLS . Finally, OLS estimation of the
transformed model
DGLS y = DGLS Xβ + DGLS W Xγ +

1
1
DGLS ΨΨ X ∗ Πµ + DGLS W ΨΨ XΠα + DGLS η,
T
T
(3.7)

provides the FGLS estimates of β, γ, Πµ and Πα .
Interestingly, we can proceed in an analogous way to deal with error structures with
idiosyncratic shocks following autoregressive and moving-average processes. In particular, the
main diﬀerence with respect to the procedure proposed to deal with spherical disturbances is
that the presence of a serial correlation matrix Kronecker (post-) multiplying the last term of
equation (3.1) results in additional (autoregressive and moving-average) parameters in (3.2).
Alternatively, we can account for the serial correlation in the model by including among the
regressors lags of the dependent variable (and possibly of the explanatory variables in X and
W X). Notice, however, that the presence of the time-invariant components υ µ and υ α in the
error term makes the lagged dependent variable endogenous. We thus propose instrumenting
this variable using lags of the explanatory variables X (and possibly W X) to control for the
endogeneity of the lagged dependent variable.

3.2

Instrumental Variables Estimation Under Sequential Exogeneity

The assumption of strict exogeneity of the covariates is critical to guarantee that the GLS
estimators presented in the previous section provide consistent estimates of the parameters of
interest. However, in applications the assumption that εit is uncorrelated with the covariates
in all the time periods may not hold. If for example the values of an explanatory variable in
period t are related to past values of the dependent variable (e.g., in t − 1), then future values
of these explanatory variables (e.g., in t + 1) may depend on the values of the idiosyncratic
term in t, thus breaking the strict exogeneity assumption (see e.g. Wooldridge 2002).
In such circumstances, a sequential exogeneity assumption, E(εit |xi1 , xi2 , . . . , xit ) = 0,

12

seems more appropriate, since it implies that present values of yit do not aﬀect present and
past values of xit . However, given the spatial structure of our model and following the strict
exogeneity case, we instead propose using an “extended sequential exogeneity assumption”
involving all the units in the sample. In mathematical terms, E(εit |xjs ) = 0 for all ∀i, j and
s ≤ t. Notice also that if (expected) future values of xit depend on yit (i.e., present values
of yit aﬀect the expected value of xit+1 ), then the explanatory variables used to construct the
correlation functions in (2.4) are endogenous by construction. In other words, the presence of
predetermined variables in X and W X means that

1
ΨΨ
T

X ∗ and

1
W ΨΨ
T

X are correlated

with the idiosyncratic term ε. Therefore, under sequential exogeneity, the GLS estimators
presented in the previous section no longer provide consistent estimates of the parameters of
interest. Rather, an IV estimator should be considered for this purpose.
The main challenge IV estimators face in practice is that it is often diﬃcult to ﬁnd good
instruments. In this case, however, the structure of the model provides natural candidates.
Namely, the means of the exogenous explanatory variables constructed using values up to
period t (rather than using all T values).11 Let


1

 1
 2


LT =  1
 3
 .
 .
.

1
T

0
1
2
1
3

.
.
.

1
T

0 ··· 0





0 ··· 0 


1
··· 0 
3

. ..
. 
.
. 
. .
.

1
1
··· T
T

be the row-standardised lower triangular matrix of ones and Γ = I N ⊗LT be the transformation
matrix that yields the backward-up-to-t mean of the variable (i.e., ΓX, for example, yields a
matrix composed by the means of the exogenous explanatory variables constructed using values
up to period t). The matrix of instruments can be thus written as Z 1 =

ΓX ΓW X

Notice that these backward means are exogenous variables under the extended sequential
exogeneity assumption. But they are also relevant, since by construction they are correlated
11

In fact, provided that the number of available periods is long enough, one may use up to lagged periods to
construct such means, i.e., one may use values up to period t − 1, t − 2, etc..

13

1
1
with the endogenous explanatory variables T ΨΨ X ∗ and T W ΨΨ X. Notice also that if we use

the same explanatory variables to construct both the instruments and the correlation functions
(or diﬀerent variables but the same number), then the model is exactly identiﬁed. However, if
all the explanatory variables are used to construct the instruments but not all the explanatory
variables are used to construct the correlation functions, then the model is overidentiﬁed.
To construct the IV estimator, we follow Hausman and Taylor (1981) and Keane and Runkle
(1992). Hausman and Taylor (1981) propose a two-step procedure to estimate linear panel data
models with endogenous explanatory variables (with respect to the idiosyncratic term, as well
as with respect to the individual eﬀect) that boils down to an initial GLS transformation of the
model (using a consistent estimate of the variance-covariance matrix of the error term) and then
an estimation of the transformed model by IV. However, Keane and Runkle (1992) show that
this procedure may not yield consistent estimates when the instruments are predetermined.
This is because the GLS-transformation proposed by Hausman and Taylor (1981) results in
individual errors that are linear combinations of the errors of the individual in all time periods.
To obtain consistent estimates, Keane and Runkle (1992) instead propose using the uppertriangular Cholesky decomposition of the serial correlation matrix (forward ﬁltering) to GLStransform the model. In essence, this is the procedure we follow, except that the complex
structure of our error term requires a diﬀerent GLS transformation and alternative orthogonality
conditions between the errors and the explanatory variables.
To be precise, we obtain the IV estimates of our model in the following way.

First,

we transform the model using the projection matrix onto the column space of the matrix
consisting of the exogenous variables and the instruments. That is, we multiply the model by
the projection matrix PZ = Z(Z Z)−1 Z , with Z =

X W X Z1

. This addresses the

endogeneity problem and makes it possible to consistently estimate the transformed model by
OLS. However, a more eﬃcient estimation may be obtained if we transform the model to obtain
ˆ
spherical disturbances. To this end, we use these OLS estimates to generate the residuals η
and, after estimating (3.6), obtain ΩIV and D IV in the same way as we did for ΩGLS and
D GLS .
In particular, since our instruments are predetermined, we propose using the upper-

14

triangular Cholesky decomposition of the inverse of the variance-covariance matrix (rather than
that of the serial correlation matrix used by Keane and Runkle 1992) to obtain D IV . However,
given the spatial structure of our model, we need to sort the data ﬁrst by time and then by
units within each time period before computing the upper-triangular Cholesky decomposition
of ΩIV .12 This guarantees that the transformed errors in period t contain elements of ηis for
s ≥ t and hence the exogeneity of our instruments in the transformed model.13
In the second step of the procedure, we estimate the GLS-transformed model by IV. This
means that we again transform the model using the projection matrix PZ , except that now the
sorting of the data requires using the matrix Γ = LT ⊗ I N to construct Z 1 . The transformed
model,
1
1
PZ D IV y = PZ D IV Xβ+PZ D IV W Xγ+ PZ D IV ΨΨ X ∗ Πµ + PZ D IV W ΨΨ XΠα +PZ D IV η,
T
T
(3.8)
is then estimated by OLS. The IV estimates of β, γ, Πµ and Πα we obtain are not only
consistent but also more eﬃcient than the initial IV estimates obtained in the ﬁrst step of the
procedure.
Lastly, it is interesting to note that, unlike the strict exogeneity case, the treatment of serial
correlation under sequential exogeneity does not follow immediately. In the present case, the
presence of lags of the idiosyncratic term ε makes any predetermined variable in the model
endogenous (not only those in the correlation functions). This is, however, not a major issue
if the number of lags is small (the order of the moving average is low) and the time dimension
12

Notice that so far we have followed the standard practice of having the data sorted ﬁrst by units
and then by time within each unit, so e.g. the dependent variable was deﬁned in Section 2 as y =
(y11 , . . . , y1T , . . . , yN 1 , . . . , yN T ) . Here we require that y = (y11 , . . . , yN 1 , y12 , . . . , yN 2 , . . . , y1T , . . . , yN T ) and
X = (x11 , . . . , xN 1 , x12 , . . . , xN 2 , . . . , x1T , . . . , xN T ) . In particular, notice that this sorting requires using
W = I T ⊗ w and Ψ = ιT ⊗ I N .
13

Notice that, in a model without spatial dependence, Keane and Runkle (1992) assume that E(ηit |z is ) = 0
for s ≤ t. However, the presence of spatially weighted covariates in the model means that, for the
proposed instruments, this only holds if the extended sequential exogeneity assumptions holds. Notice also
that the observation (i, t) of the transformed error term, D IV η, contains the original error terms ηj,t for
j = i, i + 1, i + 2, ..., N as well as ηj,s for all j and s > t. This is why, in the presence of spatial dependence,
the orthogonality condition proposed by Keane and Runkle (1992) does not suﬃce. In contrast, our extended
sequential exogeneity assumption guarantees that the proposed instruments are exogenous to the transformed
errors, since E(ηjt |z is ) = 0 for all j and s ≤ t.

15

of the panel is large. If the predetermined variable is among the regressors (X and W X),
we propose using lags of the variable as instruments; and if the predetermined variable is
in the correlation function, we similarly propose adjusting the periods used to compute the
backward means. Thus, we lose (at least) one period for each additional lagged term in the
idiosyncratic error. The problem, of course, is that if the order of the moving average process
driving the idiosyncratic term is not smaller than the number of time periods minus one, then
there is no room for using lags and backward means as instruments. In particular, by the Wold
representation theorem, this situation arises if the idiosyncratic term follows an AR process (of
any order).
Alternatively, in applications in which strict exogeneity does not hold and there is serial
correlation in the model, we may include among the regressors lags of the dependent variable
(and possibly of the explanatory variables in X and W X) and then apply the two-step
procedure previously described. Notice, however, that in this speciﬁcation the lagged dependent
variable is endogenous (for the reasons pointed out in the GLS case). We thus proposed
extending our matrix of instruments to include lags of the explanatory variables X (and possibly
W X) to control for the endogeneity of the lagged dependent variable.

4

Public capital spillovers in a production function
speciﬁcation: empirical evidence

In this section we use our correlated random eﬀects speciﬁcation and the proposed FGLS and IV
estimators to empirically address the existence of public capital spillovers. To this end, we use
a Cobb-Douglas production function speciﬁcation and yearly data from (Munnell, 1990, p. 77)
on 48 US contiguous states over the period 1970 to 1986. The output variable is the gross state
product and the inputs include public capital, private capital and labour. “The unemployment
rate is also included [in the regressions] to reﬂect the cyclical nature of productivity”. All the
variables except unemployment are in logs. This data set has the additional interest of having
been partially (e.g. Garcia-Mila et al., 1996) or totally (e.g. Baltagi and Pinnoi, 1995) used

16

in a number of studies on the relation between public capital and private output —see also
Boarnet (1998) and Sloboda and Yao (2008). In particular, some of these studies have used
spatial econometrics techniques (e.g. Holtz-Eakin and Schwartz, 1995; Kelejian and Robinson,
1997).
This accumulated evidence provides an excellent benchmark for our estimates, which were
obtained for a model speciﬁcation that uses all the inputs and their spatial counterparts to
construct the correlation functions (µ and α). Before proceeding with the estimation, however,
we considered the identiﬁcation of the proposed model. We thus computed det(X X) to ﬁnd
that it was indeed positive, which means that our identiﬁcation condition holds.
We report FGLS estimates of the model in Table 1 (coeﬃcients and variance components).
We also report the joint signiﬁcance LM-tests of each subset of coeﬃcients (β, γ, Πµ and
Πα ). The ﬁrst thing to notice is that since our model speciﬁcation uses all regressors (X and
W X) to construct the correlation functions (µ and α), estimates of β and γ reported in Table 1
correspond to the within estimates of the model. As for the coeﬃcient estimates of the variables
that compose the correlation functions, Πµ and Πα , all tend to be statistically signiﬁcant (both
individually and jointly). Lastly, all the variance components σ are statistically signiﬁcant and
have reasonable values. This supports our correlated random eﬀects model speciﬁcation. In
particular, given that we reject that Πα is zero, there is evidence of contagion in the individual
eﬀects and, given that we reject that Πµ is zero, there is evidence of correlation between the
individual eﬀects and the covariates (ﬁxed eﬀects).
[Insert Table 1]
The FGLS estimates of the β-coeﬃcients associated with the main regressors (X) are in
line with those reported in previous studies. More precisely, they are close to those reported
by Holtz-Eakin and Schwartz (1995) and Kelejian and Robinson (1997). While our estimate
of the elasticity of labour is 0.7, for example, they estimated it to be between 0.6 and 0.9;
similarly, our estimate of the elasticity of private capital is 0.2, while their estimates range from
0.06 to 0.2.14 We further concur with the lack of statistical signiﬁcance of public capital (see
14

These estimates tend to be smaller that those reported by Baltagi and Pinnoi (1995) and Garcia-Mila et al.

17

also Baltagi and Pinnoi, 1995; Garcia-Mila et al., 1996) and the statistical signiﬁcance of the
spatially weighted public capital (see the second column of Table 1, γ). Lastly, the statistical
signiﬁcance of public capital in the correlation function of the individual eﬀects (see the third
column of Table 1, Πµ ) is consistent with evidence reported by Baltagi and Pinnoi (1995, p.
396) that rejects the orthogonality between regressors and individual eﬀects “only when the
public capital stock (is) included in the production function”.
Next we explore the possibility that the explanatory variables are not exogenous but
predetermined.

Previous related studies have analysed the endogeneity of (some of) the

explanatory variables, with mixed results on the endogeneity tests (Baltagi and Pinnoi,
1995; Holtz-Eakin and Schwartz, 1995; Garcia-Mila et al., 1996) and implausible results on
the coeﬃcient estimates (Baltagi and Pinnoi, 1995; Holtz-Eakin and Schwartz, 1995). Here
we address the predeterminedness of the public capital variable (and its spatially weighted
counterpart). This would be the case, for example, if the amount states spend on public capital
is related to past values of private output (e.g., because more prosperous states are likely to
generate higher tax revenues). Under such circumstances, our previous discussion on the FGLS
estimates is ﬂawed, since the variables that compose the correlation functions deﬁning µ and
α become endogenous and the FGLS estimator is no longer consistent.
We thus report results from an IV estimation in Table 2. These were obtained using
as instruments backward-up-to-t means of all the explanatory variables and their spatially
weighted counterparts. That is, Z 1 contains the Γ−transformations of public capital, private
capital, labour and unemployment as well as of their spatially weighted counterparts.15 At
ﬁrst sight, the IV estimates of the β- and γ-coeﬃcients are not substantially diﬀerent from
those obtained by FGLS (perhaps with the exception of the public capital in γ). In contrast,
IV and FGLS estimates of the coeﬃcients associated with the variables that compose the
correlation functions, Πµ and Πα , diﬀer substantially. Indeed, a Hausman test between these
two estimators strongly rejects the null hypothesis of strict exogeneity (the statistic is 159.42).
(1996), which may suggest that ignoring spatial dependence results in overestimation of the coeﬃcients.
15

We experimented with other set of instruments (e.g., without considering unemployment and its spatial
weight) and found that coeﬃcients estimates were barely altered.

18

This supports our tenet that public capital is actually a predetermined variable.
[Insert Table 2]
We then obtained an estimate of the direct individual eﬀects (i.e., µ) and the “potential”
of their spatial spillovers (i.e., α) using these IV estimates. However, we found that while
the estimated direct eﬀfects were generally statistically signiﬁcant (both using FGLS and IV
estimates), the “potential” of the spatial spillovers of the invididual eﬀects were only statistically
signiﬁcant under the strict exogeneity assumption. Under the assumption that capital is a
predetermined variable, the estimated α’s were not statistically signiﬁcant.16
Seeking for alternative, more eﬃcient speciﬁcations that yielded statistically signiﬁcant
α’s, we used a variables-selection procedure that resulted in the speciﬁcation reported in
Table 3.17

Notice that the (common) estimated model coeﬃcients are not that diﬀerent

from those reported in Table 2. Also, the correlation between the estimated µ’s and α’s
(obtained from the estimates reported in Table 2 and Table 3) is 0.99 and 0.90, respectively.
However, 14 out of the 48 estimated α’s are now statistically signiﬁcant at standard conﬁdence
levels. Alternatively, we considered an ad-hoc model selection procedure in which we explored
diﬀerent speciﬁcations in terms of instruments (e.g., using squared terms) and/or variables
(e.g., dropping the unemployment, as in Holtz-Eakin and Schwartz 1995). This also produced
16

We use t-statistics to test the statistical signiﬁcance of the individual eﬀects and their spatial spillovers.
2
In particular, the standard errors were obtained from V ar(µ − µ) = T12 Ψ X ∗ ΣΠµ X ∗ Ψ + σµ I N −
1
2 T Ψ X ∗ Mµ E ηυ µ , with E [·] denoting the mathematical expectation, ΣΠµ = V ar(Πµ ) and Πµ −Πµ = Mµ η.
The expression for V ar(α − α) is analogous, only diﬀering in the subindices (i.e., using ΣΠα instead of ΣΠµ ,
2
2
σα instead of σµ , Mα instead of Mµ , and υ α instead of υ µ ). Under standard assumptions, these are consistent
and (approximately) normally-distributed estimators of V ar(µ − µ) and V ar(α − α), respectively. Notice that
these estimated variances allow us to also test whether the direct eﬀects of two states are statistically equal
(rather than whether the direct eﬀect of one state is statistically equal to zero, as we did).
17

In particular, we proceeded as follows. We dropped the most non-signiﬁcant variable (i.e., that with the
higher p-value) in the original speciﬁcation (that reported in Table 2), reestimated the model and determined
which was the most non-signiﬁcant variable in the new speciﬁcation. We then tested whether these two variables
were jointlty statistically signiﬁcant in the original speciﬁcation. If the null hypothesis of this Wald test was not
rejected, we dropped the two variables and considered again the most non-signiﬁcant variable in the resulting
speciﬁcation. Then, we constructed a new Wald test for the null that the three variables were not jointly
statistically signiﬁcant in the original speciﬁcation. We went on dropping variables and testing their joint
signiﬁcance untill either the jointly statistically non-signiﬁcant hypothesis was rejected or all the variables in
the model were statistically signiﬁcant.

19

speciﬁcations in which the α’s were statistically signiﬁcant and highly correlated with those
obtained using the results reported in Table 2, while the coeﬃcients of the model and their
statistical signiﬁcance remained generally unaltered. This was the case, for example, when
dropping public capital from W X, the unemployment from Πµ , and the unemployment and
its spatially weighted counterpart from the set of instruments.
Given the illustrative aim of this empirical exercise, determining which is the best model
speciﬁcation is clearly beyond the scope of this paper. It is important to stress, however, that
little diﬀerences were observed among the alternative speciﬁcations we considered when we
plotted on a map of the US states the estimated values of µ and α (available from the authors
upon request). Bearing this in mind, we have used the IV estimates reported in Table 3 to
analyse the geographical distribution of the estimated direct individual eﬀects and their spatial
spillovers. In particular, our spillover eﬀects are based on LeSage and Chih (2016), who deﬁne
“spill-in” and “spill-out” eﬀects as the cumulating oﬀ-diagonal elements from row- and columnsums of the matrix of marginal eﬀects in (2.2), respectively.
[Insert Table 3]
N

wil αj ,

To be precise, LeSage and Chih (2016) interpret the row-sums of wαj (i.e.,
l=1

assuming that wii = 0) as the “spill-in eﬀects” on the outcome of unit i. Thus, this spillin eﬀect captures the impact on the outcome of unit i of all the units neighbouring i having the
N

unobserved characteristics of j. However, we ﬁnd more interesting to report

wil αl , which is
l=1

the impact on the outcome of unit i of all the units neighbouring i having their unobserved
characteristics (i.e., the impact on the outcome of unit i of the individual eﬀects of the units
neighbouring i). This is reported in Figure 1a.
N

Similarly, LeSage and Chih (2016) interpret the colum-sums of wαj (i.e., αj

wli , assuming
l=1

that wii = 0) as “spill-out eﬀects” of unit i. In particular, this spill-out captures the impact of
unit i having the unobserved characteristics of j on the outcome of the neighbours of unit i.
N

Again, we ﬁnd more interesting to report αi

wli , which is the impact on the outcome of the
l=1

units neighbouring i of the individual eﬀect of unit i. Notice, however, that our proposed spillout eﬀect is the product of αi (the “potential” of the spatial spillovers of the individual eﬀects)

20

N

wli (which in essence determines the spatial contagion, i.e., which units are aﬀected by

and
l=1

the spillover). Since our spill-in eﬀect already reﬂects the spatial contagion of the individual
N

eﬀects, it seems more interesting to report α rather than αi

wli . This is consequently what
l=1

we do in Figure 1b.
[Insert Figure 1]
Results indicate that the geographical distribution of the spatial contagion, as measured by
the spill-in (Figure 1a) and spill-out (Figure 1b) eﬀects, follows very much the same pattern
(with some notable exceptions, such as Nebraska and Nevada). This means that most states
show spill-in and spill-out eﬀects that are of analogous magnitude. However, the “inwards”
spatial contagion of the spill-in eﬀects is generally not statistically signiﬁcant (Colorado,
Montana, Texas and Utah being the exceptions). We thus concentrate on the analysis of
the spill-out eﬀects which, as previously pointed out, tend to be statistically signiﬁcant.
Figure 1b shows that, in absolute values, West-Central (from Texas to Kansas, but also
North-Dakota and Indiana) and West-Mountain (New Mexico and Wyoming) states stand
out as the areas with the highest “outwards” spatial contagion. These are thus states with
individual eﬀects that strongly and negatively spill over the neighbouring states. Interestingly,
these spill out eﬀects are generally statistically signiﬁcant at conventional conﬁdence levels.
On the other hand, there are two areas of low spill-out eﬀects (in absolute values): the WestPaciﬁc (California and Washington) and the North-East (New York in the mid-Atlantic and
Connecticut, Rhode Island, Massachusetts and Vermont in New England). These are thus
states with individual eﬀects that negatively spill over the neighbouring states, but for which
the relative magnitude of these spillovers is small (and, in fact, not statistically signiﬁcant).
Further, Figure 1c reveals that the states that have the highest estimated values of the
direct individual eﬀects are mostly located in the North and West of the country (plus Texas
and Louisiana in the South): more precisely, in the East (Illinois, Michigan and Ohio) and West
(Nebraska) North Central, the mid-Atlantic (New York and Pennsylvania), the West-Mountain
(Montana and Wyoming) and West-Paciﬁc (California and Washington) regions. Figure 1c

21

also shows that the states with the lowest estimated values of the direct individual eﬀects
concentrate in New England (Connecticut, Maine, Massachusetts, New Hampshire, Rhode
Island and Vermont), although we also ﬁnd some states in the West-Mountain (i.e., Idaho,
Utah and Nevada) and the South-Atlantic (North and South Carolina) regions.
Lastly, it is interesting to note the overlap between Figure 1a, Figure 1b and Figure 1c. In
fact, since the estimated values of the individual eﬀects and their spatial spillovers show diﬀerent
sign (see the signs of the Π-coeﬃcients in Table 1, Table 2 and Table 3), Figure 1 points to
a negative relation between µ and α (wα). Importantly, this “proportionality” between the
direct and indirect eﬀects is not a feature of the model speciﬁcation (LeSage and Chih, 2016).
Rather, it arises as a genuine characteristic of the data.18
Therefore, states with larger/smaller estimated direct individual eﬀects tend to have
larger/smaller negative spatial spillovers (both spill-in and spill-out). Notice, however, that
this is mostly driven by New England states (with small direct individual eﬀects and small
negative spatial spillovers) and the central and southern states (with large direct individual
eﬀects and large negative spatial spillovers, except for Nebraska, which shows large direct
individual eﬀects and small negative spatial spillovers). The West and North-East states, on
the other hand, tend to have large estimated direct individual eﬀects and small negative spatial
spillovers.
To conclude, it is worth noting that negative spillovers that “might shift economic activity
from one location to another” have previously been found by, for example, Boarnet (1998, p.
382) and Sloboda and Yao (2008) with respect to the stock of and the investment in public
infrastructure, respectively. However, the source of this “crowding out” eﬀect in our model
are the unobservable characteristics of the states or “unobserved productivity” (which these
studies cannot identify). In fact, consistent with the work of Holtz-Eakin and Schwartz (1995)
and Kelejian and Robinson (1997), results reported in Table 2 show that the spatially weighted
public capital has a negative coeﬃcient, but is not statistically signiﬁcant (while private capital
18

A simple regression between µ and α, for example, indeed has a negative and statistically signiﬁcant slope
(−0.34, with a p-value of 0.03), whereas a simple regression between α and wα has a positive and statistically
signiﬁcant slope (0.86 with a p-value of 0.00).

22

is, and has a positive sign). Notice also that both private and public capital are positively
related to the states’ unobserved productivity (both variables show positive and statistically
signs in Πµ ). However, the role of these variables in the spatial spillover of the productivity,
Πα , diﬀers. While the investment in private capital is associated with negative spillovers, the
investment in public capital is associated with positive spillovers.

5

Conclusions

In this paper we analyse the problem of estimating individual eﬀects and their spatial spillovers
in linear panel data models. In particular, we consider models in which the exogenous regressors
are spatially weighted and there is no spatially lagged dependent variable (i.e., the so-called
spatial-X model). We ﬁrst show that in this model speciﬁcation the individual eﬀects and their
spatial spillovers are not identiﬁed for any spatial weight matrix. Under mild assumptions,
however, we show that they are identiﬁed in a correlated random eﬀects speciﬁcation. To
be precise, we show that there is no identiﬁcation problem in a spatial−(X, Ψ) panel data
model with correlated random eﬀects if certain rank conditions hold and the individual eﬀects
correspond to deviations with respect to the constant term.
We then consider the estimation of the parameters of the (identiﬁed) model.

Under

strict exogeneity of the covariates, OLS estimates are consistent. Here, though, we provide
more eﬃcient FGLS estimators (at least more eﬃcient with respect to the coeﬃcients of
the variables that compose the correlation functions) and propose an IV estimator to tackle
situations in which the strict exogeneity assumption may not hold and a sequential exogeneity
assumption is upheld. In particular, we suggest using the means of the exogenous explanatory
variables constructed using values up to period t as instruments for the endogenous explanatory
variables used to construct the correlation function (which, ultimately, are “means-up-to-T ”
of the exogenous variables). Also, dropping the most recent periods used to construct these
instrumental variables (i.e., using “means-up-to-(t − s)”, with s being a positive integer) may
provide further instruments and/or instruments for potentially endogenous regressors.

23

Lastly, we present results from an empirical application: the estimation of a Cobb-Douglas
production function using US state data. We ﬁnd statistically signiﬁcant diﬀerences between
the FGLS and IV estimates, which suggest that the strict exogeneity assumption that sustains
the FGLS estimates may not hold because the public capital variable is actually predetermined.
Also, IV (and FGLS) estimates show that the variables that compose the correlation functions,
as well as the variance components, all tend to be statistically signiﬁcant. This supports our
correlated random eﬀects model speciﬁcation.
The geographical distribution of the IV-estimated direct individual eﬀects and their spatial
spillovers reveals the existence of three major regions: i) Central and South states, where
both direct individual eﬀects and negative spatial spillovers tend to be large; ii) New England
states, where both the direct individual eﬀects and negative spatial spillovers tend to be small;
and iii) West and North-East states, where the estimated direct individual eﬀects tend to be
large and the negative spatial spillovers tend to be small. In addition, both the “inwards” and
“outwards” spatial contagion of the individual eﬀects (i.e., the spill-in and spill-out eﬀects)
involve negative spillovers, although this sign is mostly associated with the private capital (and
labour) and is only statistically signiﬁcant for the spill-out eﬀects. Public capital, on the other
hand, is behind the positive spatial contagion of the individual eﬀects. Consistent with previous
literature, however, public capital itself does not seem to convey statistically signiﬁcant spatial
spillovers.

24

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27

Appendix: Proofs of Propositions
Proof of Proposition 1. The model in (2.1) is not identiﬁed for any spatial weight matrix
w because Ψ and W Ψ are perfectly collinear.
det

Ψ WΨ

We prove this by showing that

is zero for any spatial weight matrix w. Let

Ψ WΨ


A=

Ψ WΨ

Ψ WΨ

=T

IN
w

w
ww




(5.1)

Then, by Schur complement,
det(A) = T 2N det (I N ) det w w − w (I N )−1 w = T 2N det (w w − w w) = 0

(5.2)

Proof of Proposition 2. Since the correlated random eﬀects spatial−(X, µ) panel data model
is linear in parameters, it is identiﬁed iﬀ det(X X) = 0. If X has full rank, it is easy to show
that det(X X) > 0 (see e.g. Corollary 14.2.14 and Theorem 14.9.4 in Harville 2008).
Proof of Proposition 3. Let λOLS , λw and λGLS be the OLS, within and GLS estimators of λ.
We prove ﬁrst that λOLS = λw . To this end, we start by noting that, by the Frisch-WaughLovell theorem and given that ΨΨ ΨΨ = T ΨΨ ,
−1

λOLS = (X M 1 X)
−1

1
with M 1 = I N T − T ΨΨ X∗ (X∗ ΨΨ X∗ )

(X M 1 y)

1
X∗ ΨΨ . Also, let Q = I N T − T ΨΨ , which satisﬁes

that Q Q = Q and QΨ = 0. Lastly, since X =

0K×1 I K×K

X M 1 = X Q. Consequently,
−1

λOLS = (X QX)

28

X Qy = λw

X∗ , it can be proved that

This concludes the ﬁrst part of the proof. Next we prove that λGLS = λOLS = λw . To
this end, we start by noting that λGLS corresponds to the OLS estimator of the considered
correlated random eﬀects model transformed using the (upper triangular part of the) Cholesky
decomposition of the inverse of the variance covariance matrix of η, Ω−1 = DD . Therefore,
by the Frisch-Waugh-Lovell theorem,
λGLS = ((D X) M 2 D X)−1 ((D X) M 2 D y)
with M 2 = I N T − D ΨΨ X∗ (X∗ ΨΨ Ω−1 ΨΨ X∗ )−1 X∗ ΨΨ D. Also, from equation (19) in
Henderson and Searle (1981),
2
Ω−1 = σε I N T + ΨΣυ Ψ

−1

1
1
I − 2 ΨΣυ Ψ Ω−1
2 NT
σε
σε
1
1
= 2 I N T − 2 Ω−1 ΨΣυ Ψ ,
σε
σε
=

which implies that ΨΣυ Ψ Ω−1 = Ω−1 ΨΣυ Ψ and, using again that ΨΨ ΨΨ = T ΨΨ , it can
be proved that
ΨΨ Ω−1 = Ω−1 ΨΨ
QΩ−1 = Ω−1 Q =

(5.3a)
1
Q
2
σε

(5.3b)

Lastly, by (5.3a) matrix M 2 can be rewritten as
M 2 = INT −

1
D ΨΨ X∗ X∗ Ω−1 ΨΨ X∗
T

−1

X∗ ΨΨ D,

and so X DM 2 = X QD. Consequently,
λGLS = X QΩ−1 X

−1

(X QΩ−1 y) = X QX

where the second equality holds by (5.3b).

29

−1

(X Qy) = λOLS = λw ,

Table 1: FGLS estimates.

Coeﬃcients

Labour
Unemployment rate
Public capital

Joint LM-test
Variance Components

Note:

γ

Πµ

Πα

0.199∗∗∗
(0.030)
0.724∗∗∗
(0.035)
−0.002
(0.001)
−0.023
(0.030)

0.260∗∗∗
(0.043)
−0.027
(0.050)
−0.007∗∗∗
(0.002)
−0.129∗∗
(0.051)

0.197∗∗∗
(0.052)
−0.212∗∗∗
(0.066)
−0.013
(0.010)
0.186∗∗∗
(0.070)

−0.477∗∗∗
(0.089)
0.101
(0.115)
0.035∗
(0.018)
0.230
(0.146)

250.07∗∗∗

17.83∗∗∗

13.10∗∗∗

8.28∗∗∗

2
σµ

2
σα

σµα

2
σε

0.0045∗∗∗
(0.0001)

Private capital

β

0.0012∗∗∗
(0.0003)

0.0017∗∗∗
(0.0001)

0.0013∗∗∗
(0.0002)

∗ p-value<0.1; ∗∗ p-value<0.05; ∗∗∗ p-value<0.01.

The dependent variable is the gross state

product. All the variables are in logs, except for the unemployment. Variance components were
estimated by OLS.

30

Table 2: IV estimates.

Coeﬃcients

Labour
Unemployment rate
Public capital

Joint LM-test
Variance Components

Note:

γ

Πµ

Πα

0.255∗∗∗
(0.037)
0.676∗∗∗
(0.059)
−0.003
(0.002)
−0.029
(0.125)

0.259∗∗∗
(0.055)
−0.045
(0.078)
−0.009∗∗∗
(0.003)
−0.100
(0.163)

0.351∗∗∗
(0.081)
−0.666∗∗∗
(0.132)
0.009
(0.015)
0.541∗∗
(0.230)

−0.601∗∗∗
(0.135)
−0.100
(0.217)
0.067∗∗
(0.029)
0.661∗
(0.347)

168.57∗∗∗

12.40∗∗∗

12.77∗∗∗

6.32∗∗∗

2
σµ

2
σα

σµα

2
σε

0.0046∗∗∗
(0.0001)

Private capital

β

0.0008∗∗∗
(0.0003)

0.0019∗∗∗
(0.0001)

0.0019∗∗∗
(0.0002)

∗ p-value<0.1; ∗∗ p-value<0.05; ∗∗∗ p-value<0.01.

The dependent variable is the gross state

product. All the variables are in logs, except for the unemployment. The matrix of instruments
consist of backward-up-to-t means of public capital, private capital, labour and unemployment as well
as of their spatially weighted counterparts. Variance components were estimated by OLS.

31

Table 3: IV estimates.

Coeﬃcients
Private Capital
Labour
Unemployment rate

β

γ

Πµ

Πα

0.252∗∗∗
(0.040)
0.666∗∗∗
(0.050)
−0.011∗∗∗
(0.002)

0.419∗∗∗
(0.068)
−0.279∗∗∗
(0.084)
−0.008∗∗∗
(0.003)

0.342∗∗∗
(0.084)
−0.776∗∗∗
(0.118)

−0.909∗∗∗
(0.180)

0.660∗∗∗
(0.132)

0.877∗∗∗
(0.193)

Public Capital

Note:

2
σµ

2
σα

σµα

2
σε

0.0044∗∗∗
(0.0001)

Variance Components

0.0026∗∗∗
(0.0003)

0.0018∗∗∗
(0.0001)

0.0015∗∗∗
(0.0002)

∗ p-value<0.1; ∗∗ p-value<0.05; ∗∗∗ p-value<0.01.

The dependent variable is the gross state

product. All the variables are in logs, except for the unemployment. The matrix of instruments
consist of backward-up-to-t means of public capital, private capital, labour and unemployment as well
as of their spatially weighted counterparts. Variance components were estimated by OLS.

32

Figure 1: Estimated individual eﬀects and their spatial spillovers.
(a) Geographical distribution of wα

(b) Geographical distribution of α

(c) Geographical distribution of µ

Note: ∗ p-value<0.1;

∗∗

p-value<0.05;

∗∗∗

p-value<0.01.