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The Mundlak approach in the spatial Durbin
panel data model
Nicolas Debarsy
CERPE, University of Namur
Rempart de la vierge 8, 5000 Namur
E-mail: [email protected]
Abstract
This paper extends the Mundlak approach to the spatial
Durbinpanel data model (SDM) to help the applied researcher to
determinethe adequacy of the random effects specification in this
setup. Wepropose a likelihood ratio (LR) test that assesses the
significance ofthe correlation between regressors and individual
effects. Once thecorrelation with individual effects has been
modeled through an aux-iliary regression, the random effects
specification provides consistentestimators and the effect of
time-constant variables can be estimated.Some Monte Carlo
simulations study the properties of this proposedLR test in small
samples and show that in some cases, it has a betterbehavior than
the Hausman test. We finally illustrate the usefulnessof the
extended Mundlak approach by estimating a house price modelwhere
some of the price determinants are time-constant. We show
thatignoring the endogeneity of regressors with respect to
individual effectsleads to unreliable estimated parameters while
results obtained usingthe Mundlak approach and the fixed effects
specification are similar(concerning time-varying variables),
implying that correlation betweenregressors and individual effects
is well captured.Keywords: Spatial autocorrelation, Panel data
model, Random ef-fects, Mundlak approach, House price modelJEL:
C12; C21; C23; C52
1 Introduction
Spatial autoregressive panel data models are of high interest
since they al-low capturing individual, temporal and interactive
heterogeneity. The firsttwo types of heterogeneity come from
individual or time characteristics andare easily dealt with using
either a fixed or random effects panel data spec-ification.
Interactive heterogeneity is due to differentiated feedback
effects,
Nicolas Debarsy is Fellow researcher of the F.R.S.-FNRS and
gratefully acknowledgestheir financial support.
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originating from cross-section interactions between
individuals.1 It cannot bedealt with traditional panel data methods
and furthermore requires explicitmodeling of spatial
autocorrelation. More precisely, interactive heterogeneityis
captured by impact coefficients or elasticities computed from the
reducedform of the spatial panel data model presented in (1),
taking into accountthe interaction structure between
individuals.
Yt = WYt +Xt +Ut, t = 1, . . . , T (1)
where Yt is the N-dimensional vector of the dependent variable
for all indi-viduals in period t, W is the spatial weight matrix
that models the inter-action scheme between individuals, Xt is the
N K matrix of explanatoryvariables and Ut is the vector of errors
possibly containing individual and/ortime effects.2 However, in
applied work LeSage & Pace (2009) advocate theuse of the
spatial Durbin model (hereafter SDM) which generalizes equation(1)
in allowing the endogenous variable to depend on the neighboring
valuesof explanatory variables (WXt), as shown in (2).
3
Yt = WYt +Xt +WXt +Ut, t = 1, . . . , T (2)
It should be clear from (2) that the SDM simplifies to the
spatial autoregres-sive model when is not significant. Also, by
imposing other constraints onthe parameters of the SDM, we can
retrieve the model with spatially autocor-related errors.4 This
spatial Durbin specification constitutes the benchmarkmodel of this
paper.
As mentioned above, individual and temporal heterogeneity are
dealtwith by estimating a fixed or random effects specification.
Anselin (1988,chap.10) is the first to suggest a one-way random
effects specification withspatially autocorrelated errors (Anselins
model hereafter). Several likeli-hood ratio (LR) and Lagrange
multiplier (LM) statistics were derived byBaltagi et al. (2003) to
assess the relevance of random individual effects andspatial
autocorrelation in this model. These statistics were further
general-ized to the presence of serial correlation by Baltagi et
al. (2007). Also, Baltagiet al. (2009b) suggest LM statistics to
test for the presence of heteroskedas-ticity and spatial
autocorrelation in Anselins model. Alternatively, Kapooret al.
(2007) (KKP hereafter) develop a method of moments estimator fora
one-way random effects model where spatial autocorrelation is
present in
1Interactive heterogeneity should not be confused with what
literature labels spatialheterogeneity and which refers to standard
individual heterogeneity coming from spatialstructural instability
in coefficients or residual variance.
2The weight matrix W and all variables should be indiced by N
since they formtriangular arrays. However, to keep some clarity in
the notations, we omitted it.
3In the context of cross-sectional models, several papers have
also shown that theSDM was theoretically grounded. Among other,
Ertur & Koch (2007, 2011) who revisitedneoclassical and
schumpeterian growth theory between countries and Pfaffermayr
(2009)who studied the convergence process among European
regions.
4For further details, see LeSage & Pace (2009, p.164).
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both components of the error term. These two different
approaches led Bal-tagi et al. (2009) to propose a generalized
random effects model that encom-passes KKP and Anselins model and
to derive LM tests to choose betweenthe two specifications.
According to Lee & Yu (2010b), this distinction isnot important
in a fixed effects specification since individual-specific
effectsenter the model as vectors of unknown fixed effects
parameters. Recently,Lee & Yu (2010a,c) have established
asymptotic properties of maximum like-lihood estimation of a
spatial autoregressive panel data model with
spatiallyautocorrelated errors with possibly both time and
individual fixed effects.
In applied work, the researcher often faces datasets where the
time spanconsidered is small compared to the number of individuals.
As such, in thispaper, we focus on the treatment of individual
effects only and time dummiescan be added to the set of explanatory
variables to account for time effects.
Even though the literature provides the estimation procedures
for bothfixed and random effects, the critical issue for the
applied researcher is todetermine the best specification for
individual heterogeneity. To this aim,Hausman (1978) derived a
statistic to evaluate the relevance of the randomeffects
specification with respect to the fixed effects model. In the
spatialpanel data literature, Mutl & Pfaffermayr (2011) and Lee
& Yu (2010b) studythe use of such a Hausman type statistic. The
former paper concentrateson the KKP specification while the
interest of the latter lies in Anselinsmodel. Finally, Lee & Yu
(2010d) also propose a Lagrange multiplier statisticbased on the
between equation to assess the relevance of the random
effectsspecification.
The originality of this paper is to suggest an alternative
testing procedureto assess the relevance of the random effects
specification. Our proceduregeneralizes Mundlak (1978) approach to
a SDM panel data model. TheMundlak approach consists in augmenting
the random effects specificationwith variables that should capture
the correlation between regressors andindividual effects. In this
paper, we suggest to go one step further and testfor the
significance of these additional variables in order first to assess
thetrade-off between bias and efficiency and second to identify the
endogenouscovariates. This last point is of high interest since it
can be used in the firststep of the methodology suggested by
Hausman & Taylor (1981). These twoauthors propose to separate
exogenous from endogenous regressors and touse functions of the
former to find consistent estimators of the latter. Orig-inally,
this distinction is based either on economic insights or some a
prioriinformation the researcher might have. The Mundlak approach
provides sta-tistical grounds to distinguish between exogenous and
endogenous explana-tory variables and would constitute a real
benefit. In that sense, Mundlaksapproach surpasses the Hausman test
since the latter only detects the pres-ence of correlation. Let us
however mention that in the Mundlak approach,the detection of
dependence between individual effects and covariates is
con-ditioned on the functional form assumed between the two while
the Hausman
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test does not rely on such an assumption. Last but not least, as
Mundlaksapproach is based on a random effects specification, it
allows to consistentlyestimate the effect of time-constant
variables even though regressors and in-dividual effects are not
independent. So, it constitutes a solution for appliedresearchers
dealing with time-constant variables but where the Hausman orLR
statistic proposed here reject the null of independence between
regres-sors and individual effects. The empirical application
proposed in this paperperfectly illustrates the last point above.
We estimate a house price modelon 588 Belgian municipalities to
assess their relevant determinants. Amongthese determinants, we
find time-constant variables that capture the attrac-tiveness of
municipalities. These explanatory variables are the results of
afactor analysis which summarizes twelve measures of attractiveness
into threefactors, each one identifying a specific aspect of
municipalities own charac-teristics, namely the charm of
environment, the quality and availability ofservices provided and
the quality of roads. As these factors are of inter-est, a random
effects specification is required since estimating this model
byfixed effects would result in the deletion of the time-constant
covariates. Weshow that ignoring the correlation between regressors
and individual leadsto unreliable estimated coefficients. However,
augmenting this random ef-fects specification with variables that
capture this endogeneity (i.e. applyinga Mundlak approach) provides
estimators similar to those obtained with afixed effects
specification (at least for time-varying covariates). Let us
fur-ther notice that as the estimated model includes an endogenous
spatial lag,the estimated coefficients are not directly
interpretable and we need to relyon the matrix of partial
derivatives associated with the reduced form of themodel to
interpret the impact of a change in an explanatory variable on
theprice of houses sold. This illustration shows that the Mundlak
approach inspatial panel data is extremely useful to estimate the
effect of time-constantvariables and furthermore allows for some
correlation between regressors andindividual effects, which is a
clear advantage over the traditional random ef-fects
specification.
The remainder of the paper is organized as follows: Section 2
describesthe model used and the testing methodology. Monte Carlo
simulations toassess the properties of the method in finite samples
are performed in Section3. Section 4 presents an empirical
illustration of the proposed methodologywhile Section 5
concludes.
2 Methodology
Our benchmark model is the one-way error component SDM presented
in(3).
Yt = WYt +Xt +WXt +Ut
Ut = +Vt, t = 1, . . . , T(3)
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Yt is the N-dimensional vector of the dependent variable in
period t, W isthe spatial weight matrix that models the interaction
scheme between theN individuals, and the spatial autoregressive
parameter to be estimated.Also, Xt is the N K matrix of explanatory
variables while WXt rep-resents the matrix of spatially lagged
explanatory variables. When W isrow-stochastic and Xt contains an
intercept, this intercept should not beincluded in the spatial lag
of the explanatory variables since WN = N .Ut is the N -dimensional
vector of errors containing both individual effects() and a vector
of innovation terms Vt = (V1,t, V2,t, . . . , VN,t)
, where Vi,tis iid across i and t with zero mean and variance 2.
Finally, as alreadymentioned, the typical dataset available to the
applied researcher containsmany individuals compared to the number
of periods of time. In case wewish to control for time effects, we
can add time dummies to equation (3).These dummies will not change
the results since they should be consideredas control variables
distinct from the covariates and their spatial lags. Forthe sake of
clarity, we will not write them explicitly in the model.
According to Mundlak (1978), the random effects specification is
a mis-specified version of the fixed effects (within) model since
it ignores the pos-sible correlation between individual effects and
regressors. By controllingfor this correlation, Mundlak shows that
coefficients of the random effectsspecification are identical to
those of the within estimation unifying in thisway both approaches.
He thus proposes to set an auxiliary regression thatwill capture
this possible relation.
Before writing down this auxiliary regression for model (3), we
rewritein (4) the model for individual i in period t. We observe
that the dependentvariable for i depends on the explanatory
variables in i but also on explana-tory variables in its
neighborhood. So, the auxiliary regression should alsocapture the
correlation between the individual effect (i) and the
neighboringcovariates.
Yi,t = Wi.Yt +Xi,t +Wi.Xt + Ui,t
Ui,t = i + Vi,t(4)
where Wi. is the ith row of W. The baseline auxiliary regression
used inthis paper is presented in equation (5). Let us note that as
i is by definitiontime-invariant, it should only be correlated with
the time-invariant part ofexplanatory variables.
i = Xi.pi +Wi.X.+ i. (5)
where Xi. = 1/TT
t=1Xi,t is the average over time of X for individual i andi.
IN(0,
2).
The independence assumption between individual effects and
regressorsis violated when, in equation (5), coefficient vectors pi
and are significant.In such a case, model (3) should include
individual fixed effects. If the null of
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non-significance cannot be rejected, individual heterogeneity in
specification(3) is best modeled by random effects. In this paper,
the significance of thesetwo vectors is assessed with a LR
statistic. This LR test over pi and is away to assess the trade-off
existing between bias and efficiency. For instance,if the results
indicate a barely significant LR statistic, the applied
researchercould wonder whether a random effects specification, even
though slightlybiased, should not be preferred to a fixed effects
model due to its gain in theprecision of the estimated parameters.
However, if the LR statistic is highlysignificant, the gain in
efficiency of the random effects does not compensatethe bias
generated and the fixed effects should be preferred.
When the LR statistic is significant, it can be of interest to
identify theendogenous regressors. To do so, we look at the
individual significance ofthe regressors used in the auxiliary
regression. This identification mightbe important when variables of
interest are time-invariant. In that case, thefixed effects
specification is useless while the Hausman-Taylors methodologycould
be helpful. In their seminal paper, these two authors propose to
splitthe regressors (both time variant and invariant) into two
sets, dependingon their relation with individual effects. the first
set contains all regres-sors considered (a priori) as exogenous
while the second include regressorsthat (a priori) could be
correlated with individual heterogeneity. They thensuggest to
instrument endogenous regressors (both varying and non-varyingover
time) by functions of time-varying exogenous covariates. The
analysisof significance of these control variables can be a first
solution to help the ap-plied researcher to determine time-varying
exogenous explanatory variableson statistical grounds.
Denote Wa = IT W, the weight matrix for the whole sample andZ =
(T IN ), a matrix of dummy variables with T a T -dimensional
vectorof ones. We can write model (3) stacked for all time periods
as follows:
Y = WaY +X +WaX +U
U = Z+V(6)
Writing the auxiliary regression set forth in (5) for all
individuals and alltime periods yields:
Z =(JT IN
)(Xpi +WaX+ )
= P(Xpi +WaX+ )(7)
with JT =T
T
Tthe operator computing averages over time. Expression
(7) comes from use of the projection matrix on the column space
of Z.Plugging this expression into (6) yields equation (8), which
is a one-wayerror component model that has to be estimated by
random effects.
Y = WaY +X +WaX +PXpi +PWaX +U
U = P+V(8)
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To estimate model (8), the standard GLS approach has to be
avoidedsince it provides biased and inconsistent estimators due to
the correlationbetween the spatially lagged dependent variable
(WaY) and the error termU. Maximum likelihood estimation procedure
for the random effects spatialpanel data model has been proposed by
Elhorst (2003, 2010) and Lee & Yu(2010b,d).
The log-likelihood function associated to equation (8) is
presented below.
lnL = NT
2ln 2pi
1
2ln ||+ T ln |IN W|
1
2U1U (9)
where U = Y WaY X WaX PXpi PWaX and is theusual variance matrix
of a one-way error component model that takes thefollowing
form:
= (T2 + 2)(
1
TT
T IN ) + 2(IT
1
TT
T IN) (10)
Lee & Yu (2010d) develop and discuss the assumptions
underlying the asymp-totic properties for the (quasi-) maximum
likelihood estimator. DefiningS() = IN W, we write the adapted
version of this set of assumptions forthe model considered as
follows:
Assumption 1. The interaction matrixW is non-stochastic,
constant through
time with diagonal elements equal to 0. This matrix is also
assumed to be
uniformly bounded in row and column sums in absolute value.
Assumption 2. Individual effects i. and the disturbances Vi,t
are inde-pendent from the regressors Z = [X, WaX, PX, WaPX]. Also,
i. IN(0, 2) and Vi,t IN(0,
2).
Assumption 3. S() is invertible for all P where P is a
compactparameter space. We also have that S()1 is uniformly bounded
in row andcolumn sums in absolute value for all P .
Assumption 4. N is large and T is finite.
Assumption 5. The regressor matrix Z has full column rank and
its ele-
ments are non-stochastic and uniformly bounded in N and T .
Also, underthe asymptotic setting of Assumption 4, the limit of
1
NTZ1Z exists and
is nonsingular. We furthermore assume that all parameters are
identified.
When the interaction matrix is row-normalized, the parameter
space P isa compact subset in (-1,1).5 The asymptotic setting
defined in Assumption4 corresponds to what is usually encountered
in applied work, meaning smallnumber of periods compared to a large
number of individuals.
The next section assesses the properties of the Mundlak approach
ex-tended to spatial Durbin panel data models through Monte Carlo
experi-ments.
5See Kelejian & Prucha (2010, p. 55) for a discussion on the
parameter space for thespatial autoregressive parameter.
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3 Monte Carlo simulations
The benchmark model for the Monte Carlo simulations is presented
in equa-tion (11):
yt = Wyt + x1,t + x2,t + 0.5Wx1,t + 0.5Wx2,t + ut
ut = + vt, t = 1, . . . , T.(11)
For these Monte Carlo experiments, two different spatial schemes
are gen-erated and both are row-normalized. In the first one, W is
defined as firstorder contiguity under the queen sense. For the
second spatial scheme, weuse a 15 nearest neighbors definition.
Observations are drawn from a squaregrid containing respectively
49, 144 and 256 observations. Furthermore, twodifferent time span
are considered: T = 5, 10. The spatial autoregressiveparameter
varies over the set [0.6, 0.6] by increment of 0.2. The
explana-tory variables x1 and x2 are formed from a time-invariant
component and anidiosyncratic term, both drawn from standard normal
distributions. In otherwords, x1i,t = 1i + 1i,t and x2i,t = 2i +
2i,t. Also, vi,t is drawn from anindependent standard normal
distribution. Finally, three configurations areexplored for the
individual effects () and summarized in Table 1. In DGP1, we
consider i as pure random effects drawn from a normal
distributionwith a standard deviation of two. In DGP 2, individual
effects are assumedto be correlated only with the time averaged
value of x1 and its spatial lagWx1. In the last DGP, individual
effects are assumed to be correlated withthe time average value of
all regressors (xi. = [x1,i., x2,i.]). The parameter captures the
intensity of correlation and takes on three different values:0.1,
0.5 and 0.8 that reflect low, medium and high dependence. Each case
isreplicated 1000 times. In Tables 2 to 4 that are discussed below,
in additionto presenting the LR test proposed in this paper, we
provide results of theHausman test developed by Lee & Yu
(2010b) and compare outcomes of bothapproaches.
Table 1: Different DGPs for individual effects
DGP 1 DGP 2 DGP 3i = i i = (x1,i. +Wx1,i.) + i i = (xi. +Wxi.) +
i
i N(0, 2) i N(0, 2) i N(0, 2)
Tables 2 to 4 present Monte Carlo results for both the LR
statistic andthe Hausman test to assess the relevance of the random
effects specification.Figures correspond to rejection rates of the
null of absence of correlationbetween regressors and individual
effects. In DGP 1, we assess the size ofboth statistics, set to a
theoretical value of 5% and highlighted in bold inthe Tables. DGP 2
and 3 study some measure of the power of the twoaforementioned
tests. All tables are split into two. The upper panel displays
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outcomes when the contiguity in the queen sense is used while
the lowerpanel makes use of the 15 nearest neighbors interaction
scheme.
Table 2 summarizes results forN = 49. We first observe that in
this smallsample case, the size of the LR statistic, even though a
bit high, is not faraway from the theoretical 5%. The Hausman test
is much more liberal butthis could be explained by the fact that it
requires much more observations tofollow a 2 distribution. Indeed,
the size of the Hausman test is closer to itsnominal value of 5%
when N = 144. In the DGP 2, where only one covariate(and its
spatial lag) is correlated with individual effects, the low
correlationcase ( = 0.1) is characterized by low power of both
tests. However, whenthe correlation grows ( = 0.5, 0.8) the
rejection rate increases. Let usnevertheless note that due to the
small sample size (N = 49) the rejectionrate does not go to 1. This
weakness is nonetheless solved when the number ofindividuals
increases. The results of DGP 3 confirm those of DGP 2 but
bothtests have a higher power when = 0.5 or 0.8. When the
correlation is weak,however, their power is small. Concerning the
time span, rejection rates arehigher when more periods are
available. This result is not surprising sincefor the LR test, more
observations can be used to compute the time-averageof regressors
that capture the possible correlation with individual
effects.Concerning the Hausman test, more observations implies
higher precision ofthe estimated parameters and thus a higher power
of the statistic.
Table 3 presents the results for N = 144. We first note that
size of bothstatistics are closer to their nominal value. Without
surprise, we observe thatfor DGP 2, the rejection rate in case of
medium or high correlation greatlyincreases. For weak correlation,
the power of the LR and Hausman statisticsare around 10%. Lastly,
results of DGP 3 indicate a slight increase in powerfor the weak
correlation case and a full rejection rate when a medium or
highcorrelation between all regressors and individual effects is
generated.
Results of the Monte Carlo experiments when N = 256 are
summarizedin Table 4. Empirical sizes (highlighted in bold)
correspond to their nominalvalue. We further observe that for both
DGP 2 and 3, the power of the LRtest (Mundlaks approach) and of the
Hausman test increases compared tothe N = 144 case.
Results for = 0.1 illustrate the trade-off between bias and
efficiency.Even though theoretically the random effects model
should be rejected,Monte Carlo experiments conclude to a low
rejection rate, indicating thatin presence of weak correlation, the
random effects specification does notperform so bad. Referring to
the Hausman statistic, it seems that coefficientestimates obtained
under the random effects and fixed effects specificationdo not
greatly differ one from the other.
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Table 2: Monte Carlo results for Mundlak and Hausman approaches,
N = 49Contiguity, N = 49, T=5
DGP 1 DGP 2 DGP 3 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5 = 0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.062 0.079 0.09 0.118 0.487 0.441 0.887
0.814 0.095 0.105 0.731 0.672 1 0.887-0.4 0.063 0.083 0.081 0.095
0.477 0.431 0.902 0.835 0.106 0.097 0.765 0.707 1 0.9-0.2 0.063
0.075 0.096 0.119 0.503 0.473 0.893 0.865 0.089 0.099 0.745 0.692 1
0.960.2 0.079 0.132 0.082 0.134 0.502 0.48 0.892 0.858 0.096 0.131
0.787 0.749 0.999 0.9930.4 0.079 0.142 0.074 0.166 0.486 0.493
0.904 0.845 0.087 0.151 0.78 0.741 1 0.9970.6 0.072 0.181 0.087
0.195 0.483 0.484 0.909 0.844 0.089 0.173 0.78 0.747 1 0.998
Contiguity, N = 49, T=10-0.6 0.067 0.052 0.076 0.064 0.489 0.379
0.917 0.863 0.093 0.082 0.996 0.969 1 0.941-0.4 0.079 0.051 0.082
0.07 0.519 0.419 0.916 0.862 0.091 0.087 0.992 0.979 1 0.963-0.2
0.075 0.064 0.1 0.078 0.511 0.434 0.919 0.885 0.09 0.085 0.992
0.985 1 0.970.2 0.076 0.097 0.102 0.109 0.521 0.505 0.925 0.89
0.107 0.138 0.989 0.981 1 0.9920.4 0.063 0.139 0.103 0.131 0.491
0.483 0.913 0.884 0.089 0.172 0.994 0.989 1 0.9880.6 0.069 0.164
0.093 0.158 0.473 0.476 0.924 0.885 0.106 0.185 0.991 0.981 1
0.978
15 nearest neigbors, N = 49, T=5 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5 =
0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.077 0.184 0.092 0.163 0.553 0.555 0.924
0.884 0.093 0.168 0.761 0.727 0.94 0.874-0.4 0.085 0.199 0.085
0.164 0.543 0.541 0.926 0.887 0.078 0.174 0.789 0.766 0.944
0.877-0.2 0.07 0.201 0.099 0.196 0.531 0.527 0.934 0.919 0.074
0.185 0.779 0.752 0.925 0.8630.2 0.074 0.246 0.101 0.234 0.55 0.571
0.936 0.9 0.08 0.214 0.765 0.755 0.936 0.8750.4 0.072 0.227 0.092
0.191 0.546 0.563 0.933 0.89 0.074 0.204 0.766 0.758 0.94 0.8810.6
0.079 0.206 0.091 0.203 0.522 0.516 0.931 0.889 0.074 0.205 0.776
0.785 0.926 0.878
Inverse distance, N = 49, T=10-0.6 0.076 0.122 0.086 0.158 0.555
0.511 0.926 0.89 0.088 0.152 0.887 0.822 0.982 0.908-0.4 0.059
0.142 0.092 0.181 0.566 0.545 0.936 0.919 0.092 0.159 0.86 0.813
0.987 0.863-0.2 0.06 0.162 0.092 0.21 0.579 0.568 0.947 0.911 0.091
0.191 0.875 0.821 0.988 0.8470.2 0.088 0.231 0.081 0.215 0.58 0.574
0.937 0.899 0.097 0.246 0.868 0.856 0.99 0.8090.4 0.067 0.22 0.072
0.205 0.56 0.519 0.935 0.921 0.078 0.222 0.861 0.838 0.988 0.790.6
0.077 0.225 0.089 0.221 0.542 0.475 0.947 0.921 0.083 0.246 0.878
0.857 0.989 0.839
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Table 3: Monte Carlo results for Mundlak and Hausman approaches,
N = 144Contiguity, N = 144, T=5
DGP 1 DGP 2 DGP 3 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5 = 0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.062 0.065 0.097 0.087 0.951 0.932 1 1
0.134 0.134 1 1 1 1-0.4 0.049 0.052 0.104 0.096 0.952 0.943 1 1
0.15 0.137 1 1 1 1-0.2 0.062 0.061 0.114 0.1 0.96 0.944 1 1 0.153
0.143 1 1 1 10.2 0.057 0.072 0.101 0.105 0.962 0.939 1 1 0.159
0.159 1 1 1 10.4 0.057 0.078 0.102 0.106 0.955 0.936 1 1 0.144
0.145 1 1 1 10.6 0.048 0.065 0.092 0.107 0.958 0.936 1 1 0.154
0.166 1 1 1 1
Contiguity, N = 144, T=10-0.6 0.059 0.048 0.112 0.088 0.964
0.948 1 1 0.141 0.114 1 1 1 1-0.4 0.06 0.057 0.1 0.08 0.95 0.939 1
1 0.157 0.131 1 1 1 1-0.2 0.061 0.052 0.105 0.1 0.948 0.929 1 1
0.14 0.124 1 1 1 10.2 0.078 0.077 0.104 0.101 0.972 0.958 1 1 0.12
0.122 1 1 1 10.4 0.056 0.065 0.096 0.104 0.963 0.949 1 1 0.143
0.139 1 1 1 10.6 0.061 0.097 0.105 0.118 0.97 0.959 1 1 0.141 0.167
1 1 1 1
15 nearest neighbors, N = 144, T=5 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5
= 0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.061 0.069 0.089 0.098 0.908 0.878 1 1
0.154 0.15 1 1 1 1-0.4 0.045 0.065 0.085 0.098 0.911 0.892 1 1
0.154 0.157 1 1 1 1-0.2 0.064 0.083 0.09 0.118 0.907 0.881 1 1
0.144 0.149 1 1 1 10.2 0.069 0.102 0.093 0.139 0.911 0.892 1 1
0.159 0.185 1 1 1 10.4 0.052 0.103 0.082 0.133 0.913 0.896 1 1
0.133 0.189 1 1 1 10.6 0.068 0.144 0.096 0.172 0.913 0.88 1 1 0.158
0.216 1 1 1 1
15 nearest neighbors, N = 144, T=10-0.6 0.055 0.059 0.099 0.087
0.954 0.931 1 1 0.16 0.138 1 1 1 1-0.4 0.054 0.059 0.093 0.112
0.953 0.936 1 1 0.153 0.153 1 1 1 1-0.2 0.054 0.074 0.077 0.108
0.957 0.948 1 1 0.139 0.144 1 1 1 10.2 0.054 0.083 0.092 0.125
0.941 0.924 1 1 0.162 0.179 1 1 1 10.4 0.052 0.105 0.092 0.127
0.945 0.928 1 1 0.163 0.18 1 1 1 10.6 0.058 0.127 0.098 0.154 0.958
0.943 1 1 0.156 0.21 1 1 1 1
11
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Table 4: Monte Carlo results for Mundlak and Hausman approaches,
N = 256Contiguity. N = 256. T=5
DGP 1 DGP 2 DGP 3 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5 = 0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.047 0.047 0.177 0.171 0.998 0.998 1 1
0.201 0.186 1 1 1 1-0.4 0.038 0.048 0.148 0.138 0.995 0.992 1 1
0.183 0.165 1 1 1 1-0.2 0.048 0.058 0.140 0.122 0.998 0.998 1 1
0.193 0.184 1 1 1 10.2 0.069 0.062 0.134 0.137 0.997 0.996 1 1
0.198 0.186 1 1 1 10.4 0.068 0.071 0.137 0.131 0.998 0.995 1 1 0.2
0.194 1 1 1 10.6 0.057 0.075 0.158 0.164 1 0.995 1 1 0.212 0.194 1
1 1 1
Contiguity. N = 256. T=10-0.6 0.054 0.055 0.131 0.127 1 0.996 1
1 0.252 0.216 1 1 1 1-0.4 0.051 0.055 0.134 0.129 1 1 1 1 0.295
0.245 1 1 1 1-0.2 0.051 0.051 0.14 0.122 0.998 0.997 1 1 0.251 0.23
1 1 1 10.2 0.054 0.053 0.136 0.133 1 1 1 1 0.254 0.229 1 1 1 10.4
0.05 0.05 0.132 0.133 1 0.998 1 1 0.248 0.233 1 1 1 10.6 0.05 0.074
0.151 0.144 1 1 1 1 0.224 0.225 1 1 1 1
15 nearest neigbors. N = 256. T=5 = 0.1 = 0.5 = 0.8 = 0.1 = 0.5
= 0.8
LR Hausman LR Hausman LR Hausman LR Hausman LR Hausman LR
Hausman LR Hausman-0.6 0.058 0.063 0.137 0.138 1 1 1 1 0.197 0.183
1 1 1 1-0.4 0.052 0.068 0.135 0.127 1 1 1 1 0.192 0.175 1 1 1 1-0.2
0.037 0.058 0.128 0.119 1 1 1 1 0.206 0.193 1 1 1 10.2 0.063 0.078
0.143 0.161 1 1 1 1 0.209 0.2 1 1 1 10.4 0.042 0.061 0.122 0.143 1
1 1 1 0.207 0.212 1 1 1 10.6 0.058 0.083 0.127 0.144 1 1 1 1 0.216
0.218 1 1 1 1
15 nearest neigbors. N = 256. T=10-0.6 0.054 0.064 0,112 0,113 1
1 1 1 0.262 0.233 1 1 1 1-0.4 0.049 0.047 0,119 0,117 1 1 1 1 0.258
0.226 1 1 1 1-0.2 0.044 0.051 0,117 0,117 1 1 1 1 0.244 0.203 1 1 1
10.2 0.055 0.058 0,123 0,123 1 1 1 1 0.23 0.232 1 1 1 10.4 0.044
0.071 0,118 0,136 1 1 1 1 0.234 0.227 1 1 1 10.6 0.053 0.084 0,115
0,145 1 1 1 1 0.239 0.241 1 1 1 1
12
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4 Empirical application
To illustrate the Mundlak approach extended to spatial Durbin
models, weestimate a spatial econometrics house price model to
explain price variationsacross 588 municipalities in Belgium over
the period 2004 to 2007.
The literature concerning house prices determination has since a
longtime recognized the importance of spatial spillovers when
modeling the de-terminants of house prices. The first papers
treating these spatial spilloversindeed date back to Can (1990,
1992) and since, many studies devoted tohouse prices models
explicitly accounted for interactions (spillovers) in suchmodels
(see among others Dubin et al. 1999, Tse 2002, Beron et al.
2004,Osland 2010). In this literature, spatial spillovers are
labeled adjacencyeffects and refer to price differentials that
cannot be justified only on thebasis of housing services. For
instance, the fact that premium householdsare willing to pay just
for the snob value of a particular location (Can &Megbolugbe
1997).
The house prices specification to be estimated in this paper
comes fromthe market clearing function produced by the interaction
of bid functions ofhouseholds and offer functions of suppliers. The
original theoretical modelhas been developed by Rosen (1974) and
extended to the presence of ad-jacency effects by Fingleton (2008).
Besides, Fingleton (2010) proposes ageneralization of the
specification set out in Fingleton (2008) to panel datamodels and
constitutes the basis of the specification we estimate in
thispaper, shown in equation (12).
Pt = WPt +X + + t, t = 1, . . . , T (12)
In this model, Pt is the vector of housing price (expressed in
logs) for allmunicipalities in period t, WPt is the endogenous
spatial lag that capturesspillovers (or adjacency effects) with W
the weight matrix modeling theinteraction scheme between
municipalities, Xt is aNK matrix of covariatesthat will be
explained below and is the K-dimensional vector of
associatedcoefficients. Finally, is the N -dimensional vector of
individual effects, whilet is the traditional idiosyncratic error
term, assumed normally distributedwith zero mean and constant
variance 2.
The dataset considered concerns 588 Belgian municipalities from
2004 to2007.6 The house prices considered are the average over each
municipality ofordinary houses sold during one year. Our house
price variable thus excludesflats, villas and development sites.7
The matrix of covariates Xt containsboth time-varying and
time-constant variables. The former includes net
6Belgium counts 589 municipalities but we were obliged to
disregard Herstappe sincethis municipality is so small that we
could not get house prices data due to confidentialityissues.
7The dataset comes from the Belgian Directorate-general
Statistics and Economic in-formation and all relevant data are
expressed in constant 2004 Euros prices.
13
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income per capita expressed in logs (lincome), which proxies for
the wealth ofa municipality and whose effect is assumed positive,
the average of the surfacesold also expressed in logs (lsurf)that
control for the size of house sold(including the land), and its
spatial lag (Wlsurf), that controls for the effectof houses size
sold in the neighboring on the house price in the
concernedmunicipality. The impact of this surface variable is also
assumed positive.The last time-varying variable included is the
density (dens) which serves as ameasure of urbanization of the
municipality and whose effect is thought to bepositive due to
competition effects between potential buyers.
Time-constantexplanatory variables include nine provinces dummies
(PRp for p = 1, . . . , 9)that capture the belonging of a
municipality to a broader geographic andadministrative entity as
well as three factors measuring the attractivenessof a
municipality. These factors result from a factor analysis with
varimaxrotation whose outcomes are presented in Table 5. The
initial 12 indicatorsthat serve the analysis are reported on the
first column of Table 5. Theseindicators were computed from the
2001 socio-economic study performed bythe National Institute of
Statistic and correspond to satisfaction indexes.8
Table 5 shows that each of the three factors determines a
specific aspectof a municipalitys attractiveness. The first factor
(env_cha) refers to thecharm of environment and gathers all
characteristic of pleasant surroundings,namely attractiveness of
buildings, air quality, calm and open space. Thesecond factor
(pri_fac) captures the presence of private facilities, meaningthe
quality and availability of shopping facilities, profession
services (doctors,hairdressers, . . .), health services and
transport facilities. The third factor(road_qua) refers to roadway
quality and is related to the quality of roads,cycle tracks and
pavements. The last column of Table 5, with the headeruniqueness is
the variance percentage of the variable not explained by
thefactors. In this application, we observe uniqueness values quite
low (close tozero), which indicates that variables are well
explained by the factors.
The measures of attractiveness considered (charming
environment,privatefacilities and roadway quality) have an interest
per-se since they allow to de-termine the municipalitys
characteristics relevant to explain house prices. Toestimate their
impact, we thus have to rely on a random effects estimationand
assume that the individual effects are independent from the
regressorsand also normally distributed with zero mean and constant
variance 2.
9
The (pseudo-)within transformation proposed by Lee & Yu
(2010a) to esti-mate a fixed effects model would wipe out the
time-invariant components ofthe regression.
In this illustration, we considered several interaction matrices
but re-ported only results concerning the contiguity based one that
has been row-
8For further details, the interested reader may consult Denil et
al. (2004).9relaxing the normality assumption is possible as long
as we estimate the model with
quasi-maximum likelihood.
14
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Table 5: Results of the factor analysisFactor 1 Factor 2 Factor
3 Uniqueness
Build. attrac. 0.96 -0.03 0.04 0.07Cleanness 0.94 0 0.21
0.08
Air 0.85 -0.28 -0.13 0.19Calm 0.85 -0.32 0.01 0.17
Open space 0.78 0.05 0.13 0.36Professions services 0.03 0.89
0.33 0.11Shopping facilities -0.25 0.87 0.2 0.14
Health services -0.07 0.79 0.3 0.28Transport facilities -0.44
0.63 0.24 0.35
Roads 0.21 0.33 0.84 0.14Cycle tracks 0.16 0.3 0.81
0.24Pavements -0.3 0.43 0.64 0.31
normalized. 10
When the assumption of independence between regressors and
individualeffects is violated, random effects estimators are biased
and inconsistent.We propose to apply the Mundlak approach proposed
in this paper to dealwith this potential endogeneity problem and we
compare the results obtainedwith this approach to a fixed effects
specification (for time-varying variables)where the correlation
between regressors and individual effects is not an issueof
concern.
Before presenting the estimation results, it is important to
note thatequation (12) is an implicit form. To assess the impact of
an explanatoryvariable on the house price variable, we first need
to compute its reducedform, shown in (13), and then the matrix of
partial derivatives.
Pt = (IN W)1(0 + 1lincomet + 2surft + 3Wsurft + 4dens)
+ (IN W)1(5env_cha+ 6pri_fac+ 7road_qua)
+ (IN W)1(
9p=1
PRp + + t) (13)
This model implies that house price level in a municipality
spill overs munici-palities. It is thus possible to assess the
impact of a change in an explanatoryvariable, the income for
instance, in municipality i on the house price levelin this
municipality i but also on house price levels in all other
municipalitiesj 6= i of the sample.
10the other interaction matrices used are a 10-nearest neighbors
and an inverse distancebased matrix with a threshold, where the
threshold took several values. All these matricesgave similar
results.
15
-
The matrix of partial derivatives of Pt with respect to the
covariate ofinterest, for instance income per capita, labeled Yt
for notational clarity, ispresented below:
YP PtYt
= 1 (IN W)1 = 1(IN + W +
2W2 + ...) (14)
When the spatial lag of the covariate is also included in the
specifica-tion, the surface sold for instance, the matrix of
partial derivatives takes thefollowing form:
SP Pt
surf t= (IN2 + 3W) (IN W)
1 (15)
where the main difference with expression (14) is the presence
of the addi-tional term 3W.
The diagonal elements of this partial derivative matrix contain
the directimpacts including own spillover effects, which are
inherently heterogeneousin presence of spatial autocorrelation due
to differentiated weights in theW matrix, whereas off-diagonal
elements represent indirect impacts. Usingobvious notations, we
have, for the impact of income per capita on houseprice levels:
Pt,iYt,i
(YP )t,ii andPt,iYt,j
(YP )t,ij (16)
The own spillover effects correspond to the feedbacks from
municipality jto i when municipality i affects j as well as longer
paths which might go frommunicipality i to j to k and back to i.
The magnitude of those direct effectsdepends on: (1) the degree of
interactions between municipalities (governedby the W matrix), (2)
the parameter measuring the strength of spatialdependence between
municipalities and (3) the estimated parameter of thecovariate of
interest, . Note also that the magnitude of pure feedback
effectsare then given by (Y
P)t,ii , where could be interpreted as representing
the direct impact of per capita income if there was no spatial
autocorrelation,i.e. if was equal to zero.
Cumulative indirects effects can be computed into two different
ways,with two complementary economic intuitions. If we want to
examine how achange of a covariate in municipality i will affect
the house price levels in allothers municipalities j 6= i, we sum
all elements but the diagonal one of theith column of the partial
derivative matrix. Economically, this interpretationcan be used to
simulate economic policy scenarii since it allows to studythe total
diffusion over space of a shock given in a specified
municipality.Alternatively, we can sum all elements excepted the
diagonal one of the ith
row of the matrix of partial derivative. By doing so, we analyze
how achange in a covariate in all municipalities j 6= i will affect
house price levelin municipality i.
16
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Table 6: Estimation results for the three methods
Dependent variable: Random effects Mundlak approach Fixed
effectslog of house price (P)
lincome 0.293 0.183 0.149(0.000) (0.000) (0.003)
lsurf 0.075 0.077 0.077(0.000) (0.000) (0.000)
Wlsurf -0.074 -0.022 -0.028(0.000) (0.136) (0.048)
density 0.002 0.018 0.016(0.000) (0.001) (0.005)
env_cha 0.053 0.056 -(0.000) (0.000)
pri_fac 0.027 0.022 -(0.000) (0.000)
road_qua 0.018 0.015 -(0.010) (0.036)
0.837 0.829 0.852(0.000) (0.000) (0.000)
Specification 62.7521 47.7362
Test (0.000) (0.000)
Figures between brackets correspond to p-values. Provinces
dummies were included inthe three specifications but were not shown
since of not interest per-se.
1 This test is the traditional Hausman statistic.2 This test is
the LR statistic mentioned above in the paper.
We finally define the average direct effect as the average of
diagonalelements of the partial derivative matrix, namely
N1tr(Y
P) when look-
ing at the impact of income per capita and the average
cumulative indi-rect effect which corresponds to the average of
columns of rows sums ofthis partial derivative matrix, cleaned of
the diagonal elements, i.e. (N
1)1N
(Y
P diag(Y
P))N .
11
Results of the estimation of equation (12) by random effects,
Mundlakapproach and fixed effects specifications are summarized in
Table 4. Weobserve that the Hausman test is significant, indicating
that the fixed effectsestimation should be preferred to the
traditional random effects model sincethe assumption of
independence between regressors and individual effectsseems
violated. We thus apply the Mundlak approach and add variables
tocapture this correlation. These additional variables are the
averages over
11LeSage & Pace (2009, chap. 2) present a comprehensive
analysis of those effects alongwith some useful summary measures in
the cross-section setting. The extension to staticpanel data models
is easily done.
17
-
time of the time-varying covariates as well as their spatial
lags. The LR testat the bottom of column 3 of Table 4 indicates
that these additional variablescapture at least part of this
correlation since the null of non-significance isstrongly
rejected.
To discuss estimation results, we rely on impacts computed from
thematrix of partial derivatives of the dependent variable with
respect to eachof explanatory variables. The averaged direct and
total indirect impacts foreach covariate and for the three
estimation techniques are reported in Table7. We also report 99%
confidence intervals for these impacts constructedfrom 10000 Monte
Carlo draws.12
12The interested reader may consult LeSage & Pace (2009,
chap. 5) for further details.
18
-
Table 7: Impacts computation for the three estimation
methods
Dependent variable: Random effects Mundlak approach Fixed
effectslog of house price (P) Average Direct Average Indirect
Average Direct Average indirect Average Direct Average Indirect
lincome 0.371 1.426 0.23 0.837 0.191 0.81(0.263) (0.471) (0.989)
(1.995) (0.093) (0.364) (0.355) (1.379) (0.056) (0.327) (0.241)
(1.481)
lsurf 0.071 -0.065 0.091 0.236 0.089 0.238(0.053) (0.09)
(-0.201) (0.064) (0.076) (0.119) (0.031) (0.442) (0.066) (0.113)
(0.01) (0.485)
density 0.003 0.26 0.023 0.084 0.02 0.086(0.002) (0.004) (0.188)
(0.350) (0.005) (0.041) (0.017) (0.152) (0.002) (0.038) (0.01)
(0.169)
env_cha 0.067 0.258 0.069 0.253 - -(0.051) (0.083) (0.187)
(0.347) (0.051) (0.087) (0.18) (0.352)
pri_fac 0.034 0.13 0.028 0.102 - -(0.018) (0.051) (0.071)
(0.208) (0.01) (0.046) (0.041) (0.178)
road_qua 0.023 0.087 0.018 0.067 - -(-0.001) (0.045) (-0.002)
(0.178) (-0.005) (0.04) (-0.017) (0.155)
Figures between brackets are the lower and upper bounds of a
confidence interval at 99% constructed using 10000 MC draws.
-
Average direct and total indirects impacts under the random
effects spec-ification (collected in the two first columns of table
7) differ from those ob-tained with the two others estimation
procedures, reinforcing the Hausmanstatistic result. For instance
the direct elasticity of income per capita onhouse price level is
0.371 compared to 0.23 and 0.191 in the Mundlak andfixed effects
specification. For the density variable, impacts for the
randomeffects specification are eight times smaller than those from
the two otherspecifications (0.003 against 0.023 and 0.02). These
results confirm that ran-dom effects estimators are not reliable
and the focus of attention will insteadbe on the interpretation of
impacts in the Mundlak specification.
Impacts computed in the Mundlak and fixed effects approaches are
reallysimilar, which implies that the correlation between
regressors and individualeffects is well captured by the auxiliary
controls. The average direct elastic-ity of income per capita on
house price is positive and significant, supportingthe results
obtained by Fingleton (2008, 2010). Hence, an increase in theincome
per capita of 10% in a municipality will increase the level of
housesprice in this municipality by 2.3%. The direct elasticity of
the surface sold isalso positive and significant, a result
consistent with the economic intuitionthat larger the house sold,
higher is the price. Results also indicate thatthe density has a
positive impact on the levels of house price. This effectcan be
explained by competition effects, reflecting a disequilibrium
betweendemand and supply of housing goods. Finally, direct impacts
for two of thethree factors measuring attractiveness of a
municipality are significant. Theenvironments charm and the
presence of private facilities have positive im-pact on the price
of houses sold. Besides, we also observe that the quality ofroads
(including cycle tracks and pavements) does not significantly
affect theprice level. House buyers are thus more affected by
environments quality,which contributes to their feeling of
well-being and the quality and availabil-ity of services present in
the municipality (avoiding frequent trips to a city)than by quality
of roads, which can be viewed as a pure technical detail ofthe
municipality.
Average total indirects impacts are also all positive and
significant exceptfor the last factor (road_qua). For instance,
increasing the income percapita of 10% in a municipality will cause
the house price levels in all othermunicipalities to increase, on
average, by 8%. Using the row interpretation,we would say that
increasing the density of one unit in all municipalitiesexcept in i
will cause the price of houses sold in i to increase by
around8.4%.
5 Conclusion
This paper extends the Mundlak approach to the spatial Durbin
model andpropose a LR test to assess the relevance of the random
effects specifica-
20
-
tion in this framework. The Monte Carlo experiments indicate
that in verysmall samples, the size of the LR test behaves much
better than the oneof Hausman test. Even though the Hausman or the
LR test concludes tothe violation of the independence between
regressors and individual effects,the Mundlak specification still
permits the estimation of time-constant vari-ables while accounting
for the endogeneity problem. Naturally, the extent towhich this
correlation is captured depends on the functional form set in
theauxiliary regression. To illustrate the usefulness of the
Mundlak approach inspatial models, we estimate a house price
regression for 588 Belgian munici-palities, where some of the
explanatory variables measure the attractivenesssof these
municipalities and are time-invariant. We first regress the
houseprice level on the set of determinants by random effects and
found out un-reliable estimators, since potential endogeneity has
been ignored. However,applying the Mundlak approach captures this
correlation between regressorsand individual effects and provides
estimators (for time-varying variables)similar to those obtained by
fixed effects. Since fixed effects estimators arenot affected at
all by this correlation, obtaining similar estimators with
theMundlak approach indicates that this methodology works quite
well. In thishouse price model, we conclude that impacts (both
directs and indirects) ofthe charm of the environment and of the
quality and availability of privateservices on the house price
levels are positive and significant while impactsof the quality of
roads is not significant. We also observe positive and sig-nificant
direct and indirect elasticities of house price with respect to
incomeper capita, surface sold and density.
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