Testing for Spatial Lag and Spatial Error Dependence in a Fixed Effects Panel Data Model Using Double Length Artificial Regressions

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Syracuse University

SURFACE

Center for Policy Research

Maxwell School of Citizenship and Public

_Affairs

Fall 9-2015

Testing for Spatial Lag and Spatial Error Dependence in a Fixed

Effects Panel Data Model Using Double Length Artificial

Regressions

Badi H. Baltagi

Syracuse University, [email protected]

Long Liu

University of Texas at San Antonio, [email protected]

Follow this and additional works at: https://surface.syr.edu/cpr

Part of the Econometrics Commons, Income Distribution Commons, and the Public Affairs, Public Policy and Public Administration Commons

Recommended Citation

Baltagi, Badi H. and Liu, Long, "Testing for Spatial Lag and Spatial Error Dependence in a Fixed Effects Panel Data Model Using Double Length Artificial Regressions" (2015). Center for Policy Research. 215.

https://surface.syr.edu/cpr/215

This Working Paper is brought to you for free and open access by the Maxwell School of Citizenship and Public Affairs at SURFACE. It has been accepted for inclusion in Center for Policy Research by an authorized administrator of

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Testing for Spatial Lag and

Spatial Error Dependence in a

Fixed Effects Panel Data Model

Using Double Length Artificial

Regressions

Badi H. Baltagi and Long Liu

Paper No. 183

September 2015

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__________

CENTER FOR POLICY RESEARCH –Fall 2015

Leonard M. Lopoo, Director

Associate Professor of Public Administration and International Affairs (PAIA)

Associate Directors

Margaret Austin

Associate Director

Budget and Administration

John Yinger

Trustee Professor of Economics and PAIA

Associate Director, Metropolitan Studies Program

SENIOR RESEARCH ASSOCIATES

Badi H. Baltagi... Economics Robert Bifulco ... PAIA Leonard Burman ... PAIA Thomas Dennison ... PAIA Alfonso Flores-Lagunes ... Economics Sarah Hamersma ...PAIA William C. Horrace ... Economics Yilin Hou ... PAIA Hugo Jales... Economics Duke Kao... Economics Jeffrey Kubik... Economics Yoonseok Lee ... Economics Amy Lutz... Sociology

Yingyi Ma ...Sociology Jerry Miner ...Economics Cynthia Morrow ... PAIA Jan Ondrich...Economics John Palmer... PAIA David Popp ... PAIA Stuart Rosenthal ...Economics Rebecca Schewe ...Sociology Amy Ellen Schwartz ... PAIA/Economics Perry Singleton………...Economics Michael Wasylenko ... ……….Economics Peter Wilcoxen...…PAIA

GRADUATE ASSOCIATES

Emily Cardon ... PAIA Brianna Carrier ... PAIA John T. Creedon... PAIA Carlos Diaz ...Economics Alex Falevich ...Economics Wancong Fu ...Economics Boqian Jiang ...Economics Yusun Kim ... PAIA Ling Li ...Economics Michelle Lofton ... PAIA Judson Murchie ... PAIA Brian Ohl... PAIA Jindong Pang ...Economics Malcolm Philogene ... PAIA William Reed ... PAIA

Laura Rodriquez-Ortiz ... PAIA Fabio Rueda De Vivero ... Economics Max Ruppersburg ...PAIA Iuliia Shybalkina ...PAIA Kelly Stevens ...PAIA Mary Stottele... PAIA Anna Swanson...PAIA Tian Tang...PAIA Saied Toossi ...PAIA Rebecca Wang ... Sociology Nicole Watson... Social Science Katie Wood ...PAIA Jinqi Ye ... Economics Pengju Zhang ...PAIA Xirui Zhang ... Economics

STAFF

Kelly Bogart...….………...….Administrative Specialist Mary Santy..….…….….……....Administrative Assistant Karen Cimilluca...….….………..…..Office Coordinator Katrina Wingle...….……..….Administrative Assistant Kathleen Nasto...Administrative Assistant

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Abstract

This paper revisits the joint and conditional Lagrange Multiplier tests derived by Debarsy

and Ertur (2010) for a fixed effects spatial lag regression model with spatial auto-regressive

error, and derives these tests using artificial Double Length Regressions (DLR). These DLR tests

and their corresponding LM tests are compared using an empirical example and a Monte Carlo

simulation.

JEL No.

C12, C21, R15

Keywords:

Double Length Regression; Spatial Lag Dependence; Spatial Error Dependence;

Artificial Regressions; Panel Data; Fixed Effects

We dedicate this paper in honor of Aman Ullah’s many contributions to econometrics. We would

like to thank two anonymous referees for their helpful comments and suggestions.

Corresponding author: Badi H. Baltagi, Department of Economics and Center for Policy

Research, 426 Eggers Hall, Syracuse, University, Syracuse, NY 13244-1020; e-mail:

[email protected]

Long L

iu: Department of Economics, College of Business, University of Texas at San Antonio,

One UTSA Circle, TX - 78249-0633; e-mail:

[email protected]

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1 Introduction

Testing

For

Spatial

Lag

and

Spatial

Error

Dependence

in

a

Fixed

Eﬀects

Panel

Data

Model

Using

Double

Length

Artiﬁcial

Regressions

∗

Badi

H. Baltagi

†

,

Long

Liu

‡

August

26,

2015

Abstract

This paper revisits the joint and conditional Lagrange Multiplier tests derived by Debarsy and Ertur (2010) for a fixed effects spatial lag regression model with spatial auto-regressive error, and derives these tests using artificial Double Length Regressions (DLR). These DLR tests and their corresponding LM tests are compared using an empirical example and a Monte Carlo simulation.

KeyWords: DoubleLengthRegression;SpatialLagDependence;SpatialErrorDependence;Artiﬁcial Regressions;PanelData; FixedEﬀects.

JELClassiﬁcation: C12, C21, R15.

Davidson and MacKinnon(1984, 1988, 1993)proposedDouble LengthRegressions (DLR) as a useful tool forderivingLMtests. TheseDLR’sarecomputationallysimpleandoutperformtheirouterproductgradient (OPG) regression counterparts. They have been applied in econometrics by Baltagi (1999) to test linear versus log-linear regressions with AR(1) disturbances. Also, Baltagi and Li (2001) to test for spatial dependenceinacross-sectionregressionmodel,andbyLeandLi(2008)whoappliedittotestforfunctional formandspatialerrordependence,tomentionafew. Recently,BaltagiandLiu(2014)extendedtheBaltagi andLi(2001)paperbyderivingtheDLRtestscorrespondingtothejointandconditionalLMtestsofspatial lag and spatial errorin a cross-section regression model. Testing forspatial dependence in a cross-section regressionhasbeenextensively studiedbyAnselin(1988a,1988b, 2001),AnselinandBera(1998),Anselin, Bera,FloraxandYoon(1996),andKr¨amer(2005),tomentionafew. Thepresenceofspatialcorrelationcan

∗_We _dedicate _this_paper _in _honor _of _Aman _Ullah’s _many _{contributions} _to _{econometrics.} _We _would _like _to _thank _two

anonymousrefereesfortheirhelpfulcommentsandsuggestions.

†_{Corresponding}_author:_Badi_H._Baltagi,_Department_of_Economics_and_Center_for_Policy_Research,₄₂₆_Eggers_Hall,_Syracuse

University,Syracuse,NY13244-1020;e-mail: [email protected].

‡_Long_Liu: _Department_of _Economics, _College _of _Business,_University _of _Texas_at _San _Antonio, _One_UTSA_Circle, _TX

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renderinferenceusingordinaryleastsquaresmisleading,seeKr¨amer(2003). Also,MynbaevandUllah(2008) who derive the asymptotic distributionof the OLS estimator in a spatial autoregressive model.1 Anselin et al. (1996) consider a spatial autoregressive cross-section regression model with spatial autoregressive disturbancesand derivea series of LagrangeMultiplier (LM)tests. Theseare called thejoint, conditional andmarginal LMtests forspatial lagandspatial errordependence. Infact,Baltagi andLi (2001)derived theDLRcounterpartforthemarginalLMtestsforspatiallagandspatialerrorconsideredbyAnselinetal. (1996)andillustratedthesetestsusingAnselin’s(1988b)empiricalexamplerelatingcrimetohousingvalues andincomefor49 neighborhoodsinColumbus, Ohioin 1980. MonteCarloexperimentsshowedthat these DLRtestshavesimilarperformancetotheirLMcounterparts.

This paper focuses on similar tests but in the context of a spatial panel data model, see Leeand Yu (2010a)andBaltagi(2011)forrecentsurveys.2 _In_fact,_Baltagi,_Song_and_Koh₍₂₀₀₃₎_derived _the_joint_LM testforspatialerrorcorrelationaswellasrandomcountryeffects. Additionally,theyderivedconditionalLM tests,whichtestforrandomcountryeffectsgiventhepresenceofspatialerrorcorrelation. Also,spatialerror correlationgiventhepresenceofrandomcountryeffects. TheseconditionalLMtestsareanalternativetothe onedirectionalLMteststhattestforrandomcountryeffectsignoringthepresenceofspatialerrorcorrelation ortheonedirectionalLMtestsforspatialerrorcorrelationignoringthepresenceofrandomcountryeffects. Baltagi and Liu (2008) derived a joint LM test which simultaneously tests for the absence of spatial lag dependenceandrandomindividualeffects. ThejointLMstatisticisthesumoftwostandard LMstatistics. Thefirstonetestsfortheabsenceofspatiallagdependenceignoringtherandomindividualeffects,andthe secondonetestsfortheabsenceofrandomindividualeffectsignoringthespatiallagdependence. Baltagiand Liu(2008)alsoderived two conditionalLM tests. Thefirstone testsfor theabsenceof randomindividual effectsallowingforthepossiblepresenceofspatiallag dependence. Thesecond onetestsfortheabsenceof spatiallag dependenceallowingforthepossiblepresenceofrandomindividual effects. DebarseyandErtur (2010) derived LM and LR tests designed to discriminate between spatially autocorrelated disturbances versus a spatially lagged dependent variable in the context of a fixed effects spatial panel data model. FollowingLeeandYu(2010b),theycombineaspatiallagmodelwithaspatiallyautocorrelateddisturbances in a fixedeffects spatial paneldatasetting. Theyderive joint,marginal as wellas conditional LMand LR tests,undertheassumptionofnormalityofthedisturbances. Theyinvestigatetheperformanceofthesetests usingMonteCarloexperiments. ThispaperderivestheDLRtestscorrespondingtothejointandconditional LMtestsforspatiallag andspatialerrorconsideredbyDebarsyand Ertur(2010). Itillustrates thesetests

1_Few_ﬁnite_sample_studies_exist_in_this_literature,_most_notably_Bao_and_Ullah₍₂₀₀₇₎_who_study_the_ﬁnite_sample_properties ofthemaximumlikelihoodestimatorinspatialmodels.

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2 The

Spatial

Dependence

Model

usinganempiricalexampleandinvestigatestheperformanceofthesetestsusingMonteCarloexperiments. The paper is organized as follows: Section 2 derives theDLR forthe presence ofspatial lagand error

]

dependence in the context of a ﬁxed eﬀects panel data model. Our suggested DLR tests and their corre-spondingLMtestsarecomparedusinganillustrativeexampleinsection3andaMonteCarlosimulationin section4. Section5 concludesthepaper.

[

Considerthespatiallagpaneldataregression model

yt= ρW yt+ Xtβ + µ + νt, t = 1, ..., T, (1)

withspatialautoregressiveremaindererrors

vt= λM νt+ ϵt, t = 1, ..., T, (2)

′

where y t= (yt1, . . . , ytN)is avectorof observationsonthedependent variablefor N regionsorhouseholds observedat time t = 1, ..., T . Xtisan N × k matrixofobservationson k explanatoryvariables. Weassume

Xt tobeoffull columnrankand itselementsareassumedto beasymptoticallyboundedinabsolutevalue.

β is a k ×1 vector of parameters. ρ and λ are scalar spatial autoregressive coeﬃcientswith |ρ| < 1 and

|λ| < 1. W and M are known N × N rownormalized spatial weightmatrices whosediagonalelements are

zero and summation of each row is 1. W and M also satisfy the conditionthat (I − λM) and (I − ρW )

′ ′

are non-singular. µ = (µ1, . . . , µN) where µi denote theith individual’s ﬁxed eﬀect. These µis are time

′

invariantrandomvariablesthatarepossiblycorrelatedwiththeexplanatoryvariables. v t= (vt1, . . . , vtN)is

a vectorofremainderdisturbancesthat followa ﬁrstorder spatialautoregressivemodel. ϵ ′t= (ϵt1, . . . , ϵtN)

and ϵti is i.i.d. over t and i andis assumed to be N(0, σ_ϵ2_). _Normality _is _needed_for _the _derivation_of _the DLR,seeDavidson andMacKinnon(1993).

¯ ¯

Deﬁne ET = IT − JT, where IT is an identity matrixof dimensionT , JT = JT/T and JT is a matrix

matrixoftheeigenvectorsofthedemeaningoperator,with FT ,T₋1the T ×(T −1)matrixofeigenvectorsof 1

√

FT ,T₋1, ιT

of ones ofdimension T , seeBaltagi (2013). Leeand Yu (2010)deﬁne as theorthonormal

T

ET correspondingto eigenvaluesequal to 1. ιT isa vectorofones ofdimension T . Deﬁnethetransformed

N ×(T −1)matrix(y∗1, . . . , y ∗T−1

)

= (y1, . . . , yT)FT ,T−1. Equivalently, thiscanbewritten as:

    ∗ y y1 . . . 1        = ( ) _      , ∗ y = .._. F_{T ,T}′ ₋₁⊗ IN ∗ y T−1 yT

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∗

where y isofdimension N (T −1)×1, IN isanidentitymatrixofdimension N,and⊗denotestheKronecker ∗ product. Onecan see that the transformedsample is shrunkbyone time period. Similarlydeﬁne X_t∗, v_t

and ϵt∗ for t = 1, ..., T −1. Thebenchmarkmodel inEquation(1)becomes:

∗ ∗

y _t = ρW y _t∗+ X _t∗β + v _t, for t = 1, ..., T −1, (3)

and

ν _t∗= λM ν _t∗+ ϵ _t∗,for t = 1, ..., T −1. (4)

Rewriteequation(3)in matrixform as

∗ ∗ ∗

y = ρ (IT₋1⊗ W )y + X ∗β + v , (5)

and

ν ∗= λ (IT₋1⊗ M) ν ∗+ ϵ ∗, (6)

where X∗ is N (T −1)×k, β is k ×1and ε∗ is N (T −1)×1. IT₋1isanidentitymatrixofdimension T −1.

( _′ ) ( _′ ) ( _′ )

Thisfollows fromthefact that F T ,T−1⊗ IN (IT ⊗ W ) = F T ,T−1⊗ W = (IT−1⊗ W ) F T ,T−1⊗ IN . A

similarresultobtainswhenwereplace W by M. Theobservationsareorderedwith t beingtheslowrunning

( )

∗′

index and i thefast runningindex, i.e., y = y11, . . . , y1N, . . . , y(T−1),1, . . . , y(T−1),N . X∗, ν∗ and ϵ∗ are

similarlydeﬁned.

Model(5)can berewrittenas

(IT₋1⊗ BA) y ∗−(IT₋1⊗ B) X ∗β = ϵ ∗, (7) where A = IN − ρW and B = IN − λM. Thisyieldsthefollowingrepresentationforthe itthobservation

∗ fit(yit, φ) = 1 ϵit∗, i = 1, . . . , N; t = 1, ..., T −1, (8) σϵ where  ₍ ₎   ′ N N N N N ∑ ∑ ∑ ∑ ∑ ∗ ∗ ∗ ∗ ∗ ∗ ∗ fit(yit, φ) = 1 _

yit− ρ wisyst− λ mijyjt+ ρλ mij wjsyst −xit− λ mijxjt β,

σϵ s=1 j=1 j=1 s=1 j=1 (9) [ ] with φ ′= β ′, σ_ϵ2_{, ρ,}_λ _. 1_ϵ_∗ Under thenormality assumption, we have _σ

ϵ it ∼N ID (0, 1) and hence thelog-likelihood function of

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N (T −1) N (T −1) L (y ∗, φ) = − ln2π − ln σ_ϵ2+ (T −1)ln|A|+ (T −1)ln|B| 2 2 1 _∗ −(IT−1⊗ B)X ∗β]′ ∗−(IT−1⊗ B) X ∗β]. − [(IT−1⊗ BA) y [(IT−1⊗ BA)y (10) 2σ_ϵ2

Ord (1975) shows that ln|A| = ln|IN − ρW | = ∑_iN₌₁ln(1− ρωi), where ωi’s are the eigenvalues of W . Similarly, ln|B| = ln|IN − λM| = ∑N_i₌₁ln(1− ληi), where ηi’s are the eigenvalues of M. As shown in Lemma 1 of Li, Yu and Bai(2013), all theeigenvalues of a row normalized spatial weight matrixare real numbers. TheJacobiantermcan berewrittenas

N (T −1) − lnσ_ϵ2+ (T −1)ln|A|+ (T −1)ln|B| 2 N N ∑ ∑ N (T −1) = (T −1) ln (1− ρωi) + (T −1) ln (1− ληi)− lnσ2ϵ (11) 2 i=1 i=1 N ∑ _∗ = (T −1) kit(y_it, φ), i=1 where ∗ kit(yit, φ)=ln (1− ρωi)+ln (1− ληi)− 1 ln σ_ϵ2, i = 1, . . . , N, t = 1, ..., T −1. (12) 2

Forthepurpose ofderivingtheDLR,thecontribution ofthe ith observationto thelog-likelihoodfunction canbe writtenas 1 1 ∗ ∗ ∗ lit(yit, φ) =− ln(2π)− f 2 it, φ) + kit(yit, φ) . (13) it(y 2 2 ∗ ∗ ∗ ∗

Deﬁning Fitj(y_it, φ) = ∂fit(y_it, φ) /∂φj and Kijt(y_it, φ) = ∂kit(y_it, φ) /∂φj, then F (y ∗, φ) and K (y ∗, φ)

∗ ∗

are matrices withtypical elements Fitj(y _it, φ) and Kitj(y _it, φ). Similarly, let f (y ∗, φ)be thevector with ∗

typicalelements fit(yit, φ).

For F (y ∗, φ),thetypicalelementsare

 ′ ∗ ∂fit(y_it, φ) ∂β ′ = N ∑ 1 _∗ _∗ − x _it− λ mijx_jt σϵ j=1 , (14) ∗ ∂fit(y_it, φ) ∂σϵ ∗ ∂fit(yit, φ) ∂ρ ∗ ∂fit(y_it, φ) ∂λ = = =  ₍ ₎  ′  N N N N N ∑ ∑ ∑ ∑ ∑ 1 _∗ _∗ _∗ _∗ _∗ _∗

− y it− ρ wisyst− λ mijyjt+ ρλ mij wjsyst −x it− λ mijxjt β,

σ2 ϵ _s₌₁ _j₌₁ _j₌₁ _s₌₁ _j₌₁  ₍ _) N N N ∑ ∑ ∑ 1 _∗ _∗ −  wisy st− λ mij wjsyst , σϵ s=1 j=1 s=1  ₍ ₎  N N N N ∑ ∑ ∑ ∑ 1 _∗ _∗ _∗′ −_σ  mijy jt− ρ mij wjsyst − mijxjtβ . ϵ _j₌₁ _j₌₁ _s₌₁ _j₌₁

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For K (y ∗, φ),thetypicalelementsare ∗ ∂kit(y _it, φ) = 0, (15) ∂β ′ ∗ ∂kit(y _it, φ) 1 = − , ∂σϵ σϵ ∗ ∂kit(y _it, φ) ωi = − , ∂ρ 1− ρωi ∗ ∂kit(y it, φ) ηi = − . ∂λ 1− ληi

TheDLRcanbe writtenasanartiﬁcialregressionwith2N (T −1)observations:

   

f (y ∗, φ) −F (y ∗, φ)

 = b +residuals, (16)

ιN(T−1) K (y ∗, φ)

where ιN(T−1)denotesavectorofonesofdimension N (T −1). ThebasicresultinDavidsonandMacKinnon (1984)is thattheinformationmatrixcan beexpressedas

[ ]

1 ( _′ _′ )

I (y ∗, φ) = plim F (y ∗, φ) F (y ∗, φ) + K (y ∗, φ) K (y ∗, φ) . (17)

N→∞ N

Anotherimportantobservation isthatthegradientofthelog-likelihoodfunctioncanbe writtenas

∂ lnL (y ∗, φ) _′ _′

g (y ∗, φ)≡ =−F (y ∗, φ) f (y ∗, φ) + K (y ∗, φ) ιN(T−1). (18)

∂φ

TheLMteststatisticinscoreformisgivenby

∗ ′ ∗ −1 _∗

g (y , φ˜) [NI (y , φ˜)] g (y , φ˜), (19)

∗

where g (y , φ˜) isthegradientevaluatedat therestrictedestimates. TheDLRvariantoftheLM testthen usesaconsistentestimateoftheinformationmatrixunderthenulloftheform

1 ( _∗ _′ _∗ _∗ _′ _∗ )

F (y , φ˜) F (y , φ˜)+ K (y , φ˜) K (y , φ˜) . (20)

N

Inthiscase,theLMtest coincideswiththeexplainedsumofsquaresfromtheDLRregressioninEquation (16). Equation(16)evaluatedat therestrictedestimatescanbewritten as

Y =Xb +residuals, (21)

   

∗ ∗

f (y , φ˜) −F (y , φ˜) ₋₁

whereY= andX = . SincetheOLSestimatorof b is(X′X) X′Y,theexplained ∗

ιN(T−1) K (y , φ˜)

−1

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[ ]

−1 ′

′ ′ ∗ ∗

theaboveartiﬁcialregressionisY I − X(X′X) X Y andY′Y = f (y , φ˜) f (y , φ˜)+ιN(T₋1)′ιN(T−1)= 2N(T −1). Therefore,theDLRteststatisticscanbealternativelycomputedas2N(T −1)minustheresidual sumofsquaresoftheaboveartiﬁcialregressions. ThisDLRtestiscomputationallysimpleandrequiresonly theeigenvaluesof W . Theseeigenvalues arealsoneededforMLestimationandtheLMtest.

2.1 Joint

DLR

test

for

H

₀a

_:

_λ

₌

_ρ

_{= 0}

Under H0a : ρ = λ =0,therestrictedMLE aretheOLS estimatesof thefollowing transformedpaneldata regression:

∗

y = X ∗β + ϵ ∗, t = 1, ..., T −1 (22)

−1

X∗′ ∗ 1 ∗′_˜∗ ∗

Inthiscase, β˜ = (X∗′X∗) y , σ˜2₌ _e_˜ _e _with_e_˜ _denoting_the_OLS_residuals_of_this_transformed

ϵ N(T−1) _[ _]

paneldata regression. Using these restrictedML estimatesφ ˜′ = β˜′, σ˜_ϵ2_{, ρ,}_˜ _λ˜ _with_ρ_˜₌_λ_{˜ =}_0,_we _run _the following2N (T −1)observationsartiﬁcialregression:

    1_e_˜∗ 1_X∗ 1 _e_˜∗ 1 ₍_I ∗ 1 ∗ T−1⊗ W ) y (IT−1⊗ M) ˜e σ˜ϵ σ˜ϵ σ˜2 ϵ σ˜ϵ σ˜ϵ  = b +residuals, (23) −1 ιN(T−1) 0 _σ_˜_ϵιN(T−1) −ιT−1⊗ ω −ιT−1⊗ η ′ ′

where ω = (ω1, . . . , ωN) aretheeigenvaluesof W, and η = (η1, . . . , ηN) aretheeigenvaluesof M. ιN(T−1) and ιT₋1 arevectorofones ofdimension N (T −1) and T −1,respectively. Theexplainedsumofsquares fromtheDLRin(23)willprovideanasymptoticallyvalidteststatisticfor H₀a_. _This_should_be_{asymptotically}

distributedas χ2 _under _the _null.3 _It _can _{alternatively}_be _computed_as ₂_N₍_T₋₁₎ _minus_the_residual_sum 2

ofsquaresoftheaboveartiﬁcialregression.4

3_In_spatial_models,_the_functional_form_changes_with_the_sample_size,_i.e.,_the_functions_F _and_K_should_have_a_sample_size subscript.Asaresult,Slutsky’sLemmanolongerappliesandreplacingthetrueparametervaluesφwithconsistentestimatesin (1)abovedoesnotnecessarilyleadtoaconsistentestimateoftheinformationmatrix.WhenFandKhavethesamefunctional formindiﬀerentsamplesizes,theircontinuitywouldguaranteethis. Here,weneedsomestrongerassumptionsonthefunctions orrequirethattheestimatesareconverginginastrongersense(e.g. almostsurely). Also,thestandardLMtestsarederived underthenormalityandhomoskedasticityassumptionsoftheregressiondisturbances.Hence,theymaynotberobustagainst non-normalityorheteroskedasticityofthedisturbances. BaltagiandYang(2013)appliedthetechniqueinBornandBreitung (2011)andintroducedgeneralmethodstomodifythestandardLMtestssothattheybecomerobustagainstheteroskedasticity andnon-normality. Thisisbeyondthescopeofthispaper,though. Weacknowledgethislimitationinthepaperandthankthe refereeforpointingitout.

4_It_is_important_to_point_out_that_the_asymptotic_distribution_of_our_test_statistics_are_not_explicitly_derived_in_the_paper. ThereisnoproofintheliteraturethattheLMtestsareasymptoticallyχ2underthenull. Giventhesimulationsbelow,they mostlikelyarebutwecannotsayunderwhatassumptionsthisholds.Thesearelikelytoholdunderasimilarsetofprimitive assumptionsdevelopedbyKelejianandPrucha(2001)fortheMoran-Itest.

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2.2 Conditional

DLR

Test

for

H

₀b

:

ρ

= 0

(allowing

an

unrestricted

estimate

of

λ

)

Under H₀b: ρ =0(allowing anunrestrictedestimateof λ),therestrictedmodelisa paneldatatransformed regressionwithﬁrstorderspatialautoregressivedisturbances:

∗ ∗

y = X ∗β + v , (24)

∗ ∗

v = λ (IT₋1⊗ M) v + ϵ ∗.

Let βˆ, λˆ andσˆ2 _denote _the_MLEs _of _β_, _λ_and _σ2 _under _this _restricted_model. _Using_these _restricted_ML

ϵ ϵ

[ ]

estimatesφ ˆ′= βˆ′, σˆ_ϵ2_,_ρ,_ˆ _λ_{ˆ with}_ρ_ˆ₌_0,_we_run_the_following₂_N₍_T₋₁₎ _observations_artiﬁcial_regression:

   ( ) ( )  1 _∗ 1 1 _∗ 1 _∗ 1 _∗ eˆ IT−1⊗ Bˆ X∗ eˆ IT−1⊗ W Bˆ y (IT−1⊗ M) ˆu σˆϵ σˆϵ σˆϵ2 σˆϵ σˆϵ  = b +residuals, ιN(T−1) 0 −_σ_ˆ1 ϵιN(T−1) −ιT−1⊗ ω −ιT−1⊗ η ∗ (25) ( ) ′ ∗ ∗ ˆ ∗ ∗ e IT−1⊗ ˆ IN λM − X∗ˆ =

where ˆ = B vˆ with B = − ˆ and vˆ = y β, ω = (ω1, . . . , ωN) and η∗

( )_′

η1 _{, . . . ,} ηN _. _The _explained _sum _of _squares _from _the _DLR _in ₍₂₅₎ _will _provide _an

asymptoti-1₋ληˆ 1 1−ληˆ N

cally valid test statistic for H₀b. This should be asymptotically distributed as χ2₁ under the null. It can alternativelybecomputedas2N (T −1)minustheresidualsumofsquaresoftheaboveartiﬁcialregression.

2.3 Conditional

DLR

Test

for

H

₀c

:

λ

= 0

(allowing

an

unrestricted

estimate

of

ρ

)

Under H₀c : λ =0 (allowing an unrestricted estimateof ρ), therestricted model is a panelspatial autore-gressiveregressionmodel oforderone:

∗ ∗

y = ρ (IT−1⊗ W )y + X ∗β + ϵ ∗. (26)

¯ _σ2

Let β, ρ¯ and ¯ denote the MLEs of β, ρ and σ2 _under _this _restricted_model. _Using _these _restricted_ML

ϵ ϵ

[ ]

estimates φ¯′= β¯′, σ¯_ϵ2, ρ,¯ λ¯ with λ¯ =0,werunthefollowing2N (T −1)observationsartiﬁcialregression:

    1_e_¯_∗ 1_X_∗ 1 _e_¯_∗ 1 ₍_IT _∗ 1 _∗ −1⊗ W ) y (IT₋1⊗ M) ¯e σ¯ϵ σ¯ϵ σ¯2 ϵ σ¯ϵ σ¯ϵ  = b +residuals, (27) −1 ιN(T−1) 0 _σ_¯ ιN(T−1) −ιT−1⊗ ω∗ −ιT₋1⊗ η ϵ ( ) ( )′ _′ ∗ ¯ ∗ ¯ ω1 ωN e IT₋1⊗ − X∗ˆ ρW , ω∗ _ρω N

where ¯ = A y β with A =IN −¯ = ₁₋_ρω_¯ , . . . , and η = (η1, . . . , ηN) .

1 1−¯

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H₀c_. _This _should _be _{asymptotically} _distributed_as_χ2 _under _the _null. _It _can _{alternatively} _be _computed _as 1

2N (T −1) minustheresidualsumofsquaresof theabove artiﬁcialregression.

3 Empirical

Illustration

Following Munnell (1990), Baltagi and Pinnoi (1995) considered the Cobb-Douglas production function relationshipinvestigatingthecontributionof diﬀerenttypesofpublicinfrastructure onprivate production. Theirregressionmodelisas follows:

log(Y ) = α + β1log(K1) + β2log(K2) + β3log(L) + β4log(U nemp) + u,

where Y is grossstateproduct, K1 ispubliccapital whichincludes highways andstreets,water and sewer facilitiesandother publicbuildingsand structures. K2 istheprivate capitalstockbasedonthe Bureauof Economic Analysis national stock estimates, L is laborinput measuredas employment in nonagricultural payrolls. U nemp is the state unemployment rate included to capture business cycle eﬀects. This panel consistsofannualobservationsfor48contiguousstatesovertheperiod1970-1986. Theweightingmatrix W =

M haselementsdiﬀerent from zeroiftwo statesareneighbors. Accordingto thequeen contiguitymatrix, Arizona and Colorado are considered neighbors. This weighting matrix has been row-normalized. Table 1 comparesthe resultsfromapplying theDLR statisticsderived inthis paper withtheir LMcounterparts derived byDebarsy and Ertur(2010). TheDLR statisticsarecomputed as 2N (T −1) minustheresidual sumof squares from (23),(25) and(27). As wecan see from the table,for allhypotheses considered, the DLRand itsLMcounterpartare closeandprovide thesame decision. Thejoint testandconditional tests rejecttheabsenceofspatialdependence.

4 Monte

Carlo

Simulation

This sectioninvestigates thesmall sample performanceof theDLRand LM tests. Following Debarsy and Ertur (2010), we generate the data using the model in Equation (1) with one regressorxit generated by

iid

xit = 0.1t + 0.5xi,t−1+ zit with zit ∼ U [−5, 5]. The trueparameters are as follows: β = 3,while ρ and λ varied over the range (−0.2, −0.5, −0.8, 0, 0.2, 0.5, 0.8). The sample size chosen is (N = 49; T = 10). The weighting matrixW = M is a row-normalized Rook-type of order 2. The individual speciﬁc eﬀects

iid iid

are µi ∼ U [−5, 5] and the disturbancesare ϵit ∼ N (0, 1). We performed1,000 replications. It is worth pointingoutthattheeigenvaluesof W needonlytobecomputedonce. Table2showsthesimulationresults. The size of thejoint DLR and LM tests for Ha _: _ρ₌ _λ₌_0, _using _the _5% _critical _value_of _a _χ2

2, is 5.8% 0

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5 Conclusion

and 5.5%, respectively. Thesimulations areperformed usinga 2.83GHz processor desktop computer with 2GB of RAM and running Windows 7 Enterprise, Service Pack 1. Computation time of the simulation resultsisshowninTable3. Forthejointtest,DLRcomputationtimein secondsismoremuchshorterthan LM. Forthe conditional tests, thecomputation time for theLM and DLRare similar. Figure 1 plotsthe correspondingpowerofthejointDLRandLMtestsfor Ha

0 : ρ = λ =0. ThesizeoftheconditionalLMtest for Hb

0 : ρ =0 (allowing anunrestrictedestimate of λ),using the5%critical valueofa χ21, variesbetween 4.8%and7.0%dependingonthevalueof λ,whilethat ofthecorrespondingDLRtest variesbetween4.7% and7.2%dependingonthevalueof λ. Figure 2plotsthecorrespondingpoweroftheconditional LMtest for Hb

0 : ρ =0 (allowing anunrestricted estimate of λ). Similarly,the size of theconditional LM test for

Hc

0 : λ =0(allowinganunrestrictedestimateof ρ),usingthe5%criticalvalueofa χ21, variesbetween3.8% and 6.3% dependingonthevalue of ρ, whilethat ofthe correspondingDLRtest varies between3.9% and 6.4%depending on thevalue of ρ. Figure 3 plotsthe correspondingpower of theconditional LM test for for H0c: λ =0 (allowinganunrestrictedestimateof ρ). As clearfrom theﬁgures,thejointandconditional DLRandLMtestsyieldsimilarsmallsampleperformanceintheMonteCarloexperiments.

This paperderivesthree artificial DLRtestscorresponding tothe LMtests derivedby Debarsy andErtur (2010) for the fixed effects spatial lag regression model with spatial auto-regressiveerror. The first DLR jointly tests for zero spatial lag dependence as well as zero spatial autoregressive error in a fixed effects paneldata model. Thesecond DLRconditionallytests forzerospatial lagdependence allowingforspatial error dependence. Whilethe third DLR conditionally tests for zero spatial error dependence allowingfor spatiallag dependence. Theproposedtests areillustratedusing theproductivitypuzzleempiricalexample byMunnell(1990). Inaddition,MonteCarloexperimentsshowthatthesmallsampleperformanceofthese DLRtestshavesimilarperformancetotheircorrespondingLMcounterparts. Furthermore,itwouldbenice ifthenormalityassumptionofthedisturbancescanberelaxed,thoughthisisbeyondthescopeofthispaper.

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Table1: Empirical TestResults Statistic p-value Ha 0 : λ ρ =0 LMλρ 243.405 0.000 DLRλρ 191.157 0.000

H0b: ρ =0 (allowinganunrestrictedestimateof λ)

LMρ_|λ 5.960 0.015 DLRρ_|λ 6.133 0.013

Hc

0: λ =0 (allowinganunrestrictedestimateof ρ)

LMλ|ρ 34.326 0.000 DLRλ|ρ 34.495 0.000

Figure 1: Testfor Ha

0 : λ ρ =0 rho −0.5 0.0 0.5 lambda −0.5 0.0 0.5 P_o w er 0.2 0.4 0.6 0.8 1.0 rho −0.5 0.0 0.5 lambda −0.5 0.0 0.5 P o_w er 0.2 0.4 0.6 0.8 1.0 = =

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Table2: SimulationResults ρ λ LMλρ DLRλρ LMρ_|λ DLRρ_|λ LMλ_|ρ DLRλ_|ρ -0.8 -0.8 1.000 1.000 1.000 1.000 1.000 1.000 -0.5 1.000 1.000 1.000 1.000 0.953 0.961 -0.2 1.000 1.000 1.000 1.000 0.338 0.369 0 1.000 1.000 1.000 1.000 0.038 0.039 0.2 1.000 1.000 1.000 1.000 0.406 0.378 0.5 1.000 1.000 1.000 1.000 0.994 0.994 0.8 1.000 1.000 1.000 1.000 1.000 1.000 -0.5 -0.8 1.000 1.000 1.000 1.000 1.000 1.000 -0.5 1.000 1.000 1.000 1.000 0.953 0.960 -0.2 1.000 1.000 1.000 1.000 0.319 0.354 0 1.000 1.000 1.000 1.000 0.045 0.046 0.2 1.000 1.000 1.000 1.000 0.432 0.405 0.5 0.999 0.999 1.000 1.000 0.994 0.994 0.8 1.000 1.000 0.999 0.999 1.000 1.000 -0.2 -0.8 1.000 1.000 0.881 0.888 1.000 1.000 -0.5 1.000 1.000 0.839 0.852 0.964 0.971 -0.2 0.994 0.999 0.803 0.808 0.288 0.327 0 0.797 0.825 0.724 0.726 0.056 0.060 0.2 0.586 0.584 0.689 0.692 0.427 0.407 0.5 0.988 0.985 0.574 0.575 0.989 0.988 0.8 1.000 1.000 0.520 0.522 1.000 1.000 0 -0.8 1.000 1.000 0.060 0.060 1.000 1.000 -0.5 0.993 0.995 0.049 0.049 0.971 0.975 -0.2 0.362 0.418 0.049 0.049 0.344 0.382 0 0.058 0.055 0.048 0.049 0.063 0.064 0.2 0.569 0.523 0.049 0.049 0.435 0.408 0.5 0.999 0.999 0.070 0.072 0.992 0.989 0.8 1.000 1.000 0.049 0.047 1.000 1.000 0.2 -0.8 1.000 1.000 0.922 0.920 1.000 1.000 -0.5 0.934 0.946 0.869 0.869 0.969 0.978 -0.2 0.754 0.754 0.808 0.813 0.362 0.388 0 0.909 0.898 0.741 0.747 0.047 0.048 0.2 0.996 0.996 0.672 0.678 0.406 0.376 0.5 1.000 1.000 0.568 0.567 0.991 0.990 0.8 1.000 1.000 0.575 0.570 1.000 1.000 0.5 -0.8 1.000 1.000 1.000 1.000 1.000 1.000 -0.5 1.000 1.000 1.000 1.000 0.976 0.979 -0.2 1.000 1.000 1.000 1.000 0.351 0.377 0 1.000 1.000 1.000 1.000 0.046 0.041 0.2 1.000 1.000 1.000 1.000 0.417 0.396 0.5 1.000 1.000 0.997 0.998 0.989 0.988 0.8 1.000 1.000 1.000 1.000 1.000 1.000 0.8 -0.8 1.000 1.000 0.999 0.997 1.000 1.000 -0.5 1.000 1.000 1.000 1.000 0.989 0.990 -0.2 1.000 1.000 1.000 1.000 0.379 0.396 0 1.000 1.000 1.000 1.000 0.040 0.042 0.2 1.000 1.000 1.000 1.000 0.454 0.434 0.5 1.000 1.000 1.000 1.000 0.993 0.992 0.8 1.000 1.000 1.000 1.000 1.000 1.000

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Table3: CompuationTimeofSimulationResults ρ λ LMλρ DLRλρ LMρ_|λ DLRρ_|λ LMλ_|ρ DLRλ_|ρ -0.8 -0.8 154.60 41.94 542.95 539.57 673.19 675.59 -0.5 148.48 30.53 544.75 542.49 674.80 675.94 -0.2 148.80 30.47 544.40 542.10 674.16 678.12 0 148.25 30.30 544.70 541.72 673.82 675.95 0.2 148.26 30.59 545.35 543.84 674.69 676.14 0.5 147.86 30.64 544.80 542.94 676.17 676.93 0.8 148.26 30.64 544.74 542.79 674.67 677.79 -0.5 -0.8 148.47 30.68 546.33 542.01 675.76 679.71 -0.5 148.74 30.84 547.64 544.32 673.79 676.17 -0.2 148.17 30.51 544.94 541.81 673.86 677.78 0 148.08 30.44 544.64 542.35 674.36 676.28 0.2 148.40 30.57 546.45 541.84 673.88 676.43 0.5 196.23 30.67 545.19 541.97 684.14 687.69 0.8 150.58 31.00 552.47 550.78 679.82 687.33 -0.2 -0.8 148.87 31.25 548.63 544.34 677.87 693.14 -0.5 149.06 30.53 546.89 545.52 675.74 678.69 -0.2 148.59 30.77 546.10 543.35 677.67 678.02 0 148.48 30.53 546.45 543.23 675.36 678.24 0.2 148.53 31.41 546.88 561.44 674.94 677.40 0.5 149.51 30.52 545.38 542.27 674.32 676.56 0.8 148.40 30.65 545.27 566.47 744.56 677.72 0 -0.8 148.16 30.32 545.14 542.01 674.10 676.78 -0.5 148.28 30.53 546.05 541.98 674.27 676.84 -0.2 148.44 30.34 545.13 542.47 675.45 676.98 0 148.17 30.31 545.08 541.71 673.98 677.48 0.2 148.16 30.29 544.86 541.79 674.01 676.61 0.5 148.19 30.52 546.13 542.79 674.62 676.86 0.8 148.41 30.33 545.12 542.29 674.92 676.40 0.2 -0.8 148.61 30.69 545.20 542.12 674.49 677.77 -0.5 157.19 43.37 555.58 547.00 682.05 684.23 -0.2 150.21 31.62 552.50 560.22 698.88 731.89 0 275.49 30.71 545.99 541.87 682.82 687.41 0.2 150.31 32.19 563.30 553.04 685.14 680.04 0.5 149.47 31.19 554.56 571.71 738.19 715.10 0.8 148.28 30.83 542.95 543.64 675.61 677.32 0.5 -0.8 147.90 31.00 543.07 543.73 676.69 679.20 -0.5 148.28 30.84 554.07 555.43 686.90 678.99 -0.2 147.67 30.96 542.35 541.09 673.85 675.77 0 147.46 30.66 541.37 541.43 675.33 676.45 0.2 148.01 30.93 542.52 542.56 675.55 677.72 0.5 148.01 30.85 542.45 542.84 675.54 676.97 0.8 147.95 30.98 542.45 543.11 675.06 676.42 0.8 -0.8 147.82 30.89 542.15 542.72 679.29 693.55 -0.5 151.58 31.00 542.62 542.97 675.73 677.56 -0.2 148.22 30.90 547.90 550.36 685.30 678.20 0 147.76 31.06 544.35 544.20 677.93 676.78 0.2 148.03 30.94 543.32 543.01 675.34 677.74 0.5 148.01 31.01 542.88 542.76 675.22 677.35 0.8 148.16 30.87 544.19 544.46 676.66 678.98 15

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Figure 2: Testfor Hb

0: ρ =0 (allowinganunrestrictedestimateof λ)

0.0 0.2 0.4 0.6 0.8 1.0 ρ P o w er −0.9 −0.6 −0.3 0.0 0.2 0.4 0.6 0.8 λ =−0.8 λ =−0.5 λ =−0.2 λ =0 λ =0.2 λ =0.5 λ =0.8 (a)LM Test 0.0 0.2 0.4 0.6 0.8 1.0 ρ P o w er −0.9 −0.6 −0.3 0.0 0.2 0.4 0.6 0.8 λ =−0.8 λ =−0.5 λ =−0.2 λ =0 λ =0.2 λ =0.5 λ =0.8 (b)DLR Test

Figure 3: Testfor H0c: λ =0 (allowinganunrestrictedestimateof ρ)

0.0 0.2 0.4 0.6 0.8 1.0 λ P o w er −0.9 −0.6 −0.3 0.0 0.2 0.4 0.6 0.8 ρ =−0.8 ρ =−0.5 ρ =−0.2 ρ =0 ρ =0.2 ρ =0.5 ρ =0.8 0.0 0.2 0.4 0.6 0.8 1.0 λ P o w er −0.9 −0.6 −0.3 0.0 0.2 0.4 0.6 0.8 ρ =−0.8 ρ =−0.5 ρ =−0.2 ρ =0 ρ =0.2 ρ =0.5 ρ =0.8