Specification tests for spatial panel data models · in spatial model speciﬁcation search means...

ORIGINAL PAPER

Specification tests for spatial panel data models

Anil K. Bera1 • Osman Dogan1 • Suleyman Taspınar2 • Monalisa Sen1

Received: 10 April 2020 / Accepted: 13 July 2020 / Published online: 30 July 2020� Springer Nature Switzerland AG 2020

AbstractSpecification of a model is one of the most fundamental problems in econometrics.

In practice, specification tests are generally carried out in a piecemeal fashion, for

example, testing the presence of one-effect at a time ignoring the potential presence

of other forms of misspecification. Many of the suggested tests in the literature

require estimation of complex models and even then those tests cannot account for

multiple forms of departures from the model under the null hypothesis. Using Bera

and Yoon (Econom Theory 9(04):649–658, 1993) general test principle and a spatial

panel model framework, we first propose an overall test for ‘‘all’’ possible mis-

specification. Then, we derive adjusted Rao’s score tests for random effect, serial

correlation, spatial lag and spatial error, which can identify the definite cause(s) of

rejection of the basic model and thus aiding in the steps for model revision. For

empirical researchers, our suggested procedures provide simple strategies for model

specification search employing only the ordinary least squares residuals from a

standard linear panel regression. Through an extensive simulation study, we eval-

uate the finite sample performance of our suggested tests and some of the existing

procedures. We find that our proposed tests have good finite sample properties both

in terms of size and power. Finally, to illustrate the usefulness of our procedures, we

provide an empirical application of our test strategy in the context of the conver-

gence theory of incomes of different economies, which is a widely studied empirical

problem in macro-economic growth theory. Our empirical illustration reveals the

problems in using and interpreting unadjusted tests, and demonstrates how these

problems are rectified in using our proposed adjusted tests.

Keywords Rao’s score (RS) test � Robust RS test � Specification test � Spatialmodels � Spatial panel data models

JEL Classification C13 � C21 � C31

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s43071-

020-00003-y) contains supplementary material, which is available to authorized users.

Extended author information available on the last page of the article

123

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1 Introduction

Econometricians’ interest in problems that arise when the assumed model that is

used in constructing a test deviates from the data generating process (DGP) goes a

long way back. As emphasized by Haavelmo (1944), in testing any economic

relations, specification of a set of possible alternatives, called the priori admissible

hypothesis, X0, is of fundamental importance. Misspecification of the priori

admissible hypotheses was referred to as Type-III error by Bera and Yoon (1993),

and Welsh (2011, p. 119) also pointed out a similar concept in the statistics

literature. Broadly speaking, the alternative hypothesis may be misspecified in threedifferent ways. In the first one, what we shall call ‘‘complete misspecification’’, the

set of assumed alternative hypothesis, X0, and the DGP X0, say, are mutually

exclusive. This happens, for instance, if in the context of panel data model, one test

for serial independence when the DGP has random individual effects but no serial

dependence. The second case, ‘‘underspecification’’ occurs when the alternative is a

subset of a more general model representing the DGP, i.e., X0 � X0. This happens,for example, when both serial correlation and individual effects are present, but are

tested separately (one at a time assuming absence of other effect). The last case is

‘‘overtesting’’ which results from overspecification, i.e., when X0 � X0. This can

happen if a joint test for serial correlation and random individual effects is

conducted when only one effect is present in the DGP. It can be expected that

consequences of overtesting may not be that serious (possibly will only lead to some

loss of power), whereas those of undertesting can lead to highly misleading results,

seriously affecting both size and power (see Bera and Jarque 1982; Bera 2000).

Using the asymptotic distributions of standard Rao’s score (RS) test under local

misspecification, Bera and Yoon (1993) suggested an adjusted RS test that is robust

under misspecification and asymptotically equivalent to the optimal Neyman

(1959)’s CðaÞ test. As we will discuss, an attractive feature of this approach is that

the adjusted test is based on the joint null hypothesis of no misspecification, thereby

requiring estimation of the model in its simplest form. A surprising additivity

property also enables us to calculate the adjusted tests quite effortlessly.

The plan of the rest of the paper is as follows. In Sect. 2, we provide a brief

review of existing literature on testing panel data and spatial models. We develop

the spatial panel data model framework in Sect. 3 and present the log-likelihood

function along with assumptions needed for asymptotic theory. Section 4 presents

the main results on the general theory of tests when the alternative model is

misspecified and then formulates the adjusted diagnostic tests which take account of

misspecification in multiple directions. To illustrate the usefulness of our proposed

tests, in Sect. 5, using an empirical example, we demonstrate how our methodology

can assist a practitioner to reformulate his/her model. For that purpose, we use

Heston et al. (2002) Penn World Table that contains data on real income,

investment and population (among many other variables) for a large number of

countries and the growth-model of Ertur and Koch (2007). From our illustration, it

is clear that the use of the unadjusted RS tests can lead to misleading inference

whereas the suggested adjusted versions of the tests lead to right direction of

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3 of 39 A. K. Bera et al.

specification search. To investigate the finite sample performance of our suggested

and some available tests, we carry out an extensive simulation study, and the results

are reported in Sect. 6. Finally, we conclude in Sect. 7. Some of technical results are

provided in an online appendix.

2 A brief survey of the literature

The origins of specification testing for spatial models can be traced back to Moran

(1950). Much later this area was further enriched by many researchers, for example,

see Anselin (1988, 2001), Anselin and Bera (1998), Baltagi and Zhenlin (2013),

Benjamin and Jorg (2011), Brandsma and Ketellapper (1979a, (1979b), Burridge

(1980), Cliff and Ord (1972), Kelejian and Prucha (2001), Kelejian and Robinson

(1992), Liu and Prucha (2018), Robinson and Francesca (2014) and Yang (2015).

Most of these papers focused on tests for specific alternative hypothesis in the form

of either spatial lag or spatial error dependence based on ordinary least squares

(OLS) residuals. Separate applications of one-directional tests when other or both

kinds of dependencies are present will lead to unreliable inference. It may be natural

to consider a joint test for lag and error auto-correlations. Apart from the problem of

over-testing (when only one kind of dependence characterizes the DGP), the

problem with such a test is that we cannot identify the exact nature of spatial

dependence once the joint null hypothesis is rejected. One approach to deal with this

problem is to use the conditional tests, i.e., to use test for spatial error dependence

after estimating a spatial lag model, and vice versa. This, however, requires the

maximum likelihood (ML) estimation, and the simplicity of test based on OLS

residuals is lost.

Anselin et al. (1996) was possibly the first paper to study systematically the

consequences of testing one kind of dependence (lag or error) at a time. Using the Bera

and Yoon (1993) general approach, Anselin et al. (1996) developed OLS-based

adjusted RS test for lag (error dependence) in the possible presence of error (lag)

dependence. Their Monte Carlo study demonstrated that the adjusted tests are very

capable of identifying the exact source(s) of dependence and they have very goodfinite

sample size and power properties. In a similar fashion, in context of panel data model,

Bera et al. (2001) showed that when we test for either random effects or serial

correlation without taking account of the presence of other effect, the test rejects the

true null hypothesis far too often under the presence of the unconsidered parameter.

They found that the presence of serial correlation made the Breusch and Pagan (1980)

test for random effects to have excessive size. Similar over-rejection occurs for the test

of serial correlation when the presence of random effect is ignored. Bera et al. (2001)

developed size-robust tests (for random effect and serial correlation) that allow

distinguishing the source(s) of misspecification in specific direction(s).

Now if we combine the models considered in Anselin et al. (1996) and Bera et al.

(2001), we have the spatial panel data model, potentially with four sources of departures(from the classical regressionmodel) coming from the possible presence of the spatial lag,

spatial error, random effect and (time series) serial correlation. The spatial panel data

models have been studied extensively in terms of estimation issues, and have gainedmuch

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Specification tests for spatial panel data models of 39 3

popularity over time given the wide availability of the longitudinal data [see, for instance,

Aquaro et al. (2019), Baltagi et al. (2014), Elhorst et al. (2014), Kapoor et al. (2007), Lee

andYu (2010), LeSage (2014), Li (2017), Olivier and LeSage (2011), Olivier and LeSage

(2012), HashemandElisa (2011), Zhenlin (2018) andYu et al. (2008)].Many researchers

have conducted conditional and marginal specification tests in spatial panel data models.

Baltagi et al. (2003) proposed conditional Lagrange multiplier (LM) tests, which test for

random regional effects given the presence of spatial error correlation and also, spatial

error correlation given the presence of random regional effects. Baltagi et al. (2007) add

another dimension to the correlation in the error structure, namely, serial correlation in the

remainder error term. Both these were based on the extension of spatial error models

(SEM). Baltagi and Liu (2008) developed similar LM and likelihood ratio (LR) tests with

spatial lag dependence and random individual effects in a panel data regression model.

Their paper derives conditional LM tests for the absence of random individual effects

under the possible presence of spatial lag dependence and vice-versa. Baltagi et al. (2009)

considered a panel data regression with heteroscedasticity as well as spatially correlated

disturbances. As in previous works, Baltagi et al. (2009) derived the conditional LM and

marginal LM tests.

For a static spatial panel data model that has individual fixed effects, Debarsy and

Ertur (2010) develop several LM test statistics as well as their likelihood ratio (LR)

counterparts for testing spatial dependence. Based on Bera and Yoon (1993), Montes-

Rojas (2010) has proposed an adjusted RS test for autocorrelation in presence of

random effects and vice-versa, after estimating the spatial dependent parameter using

ML and instrumental variable estimation methods. Similar adjusted tests are suggested

by Taspınar et al. (2017) for a higher order spatial dynamic panel data model in a

generalized method of moments (GMM) framework. However, the specification tests

proposed in the above papers require the ML estimation of nuisance parameters [except

for the adjusted tests in Taspınar et al. (2017)], and such a strategy will get more

complex as we add more parameters to generalize the model in multiple directions.

In this paper, we investigate a number of strategies to test against multiple forms

of misspecification in a spatial panel data model. We derive an overall test and a

number of adjusted tests that take the account of possible misspecification in

multiple directions. For empirical researchers our suggested procedures provide

simple strategies to identify specific direction(s) in which the basic model needs

revision using only OLS residuals from the standard linear model for spatial panel

data. The possibility of using only OLS estimator to construct RS-type adjusted tests

in spatial model specification search means wide applicability of our suggested

procedure in empirical research.

3 A spatial panel data model

We consider the following spatial panel model1:

1 Note that the variables in spatial models are allowed to depend on the number of cross-sectional units

N to form triangular arrays (Kelejian and Prucha 2010). We suppress the subscript N in stating our model

for notational simplicity.

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yit ¼ s0XN

j¼1

mijyjt þ X0itb0 þ uit; ð3:1aÞ

uit ¼ li þ �it; where li � IIDNð0; r2l0Þ; ð3:1bÞ

�it ¼ k0XN

j¼1

wij�jt þ vit; ð3:1cÞ

vit ¼ q0vit�1 þ eit; where eit � IIDNð0; r2e0Þ; ð3:1dÞ

for i ¼ 1; 2; . . .;N and t ¼ 1; 2; . . .; T . Here, yit is the observed value of the

dependent variable for location/unit i at time period t, Xit denotes the k � 1 vector of

observations on non-stochastic regressors, and eit is the regression disturbance

term.2 Spatial dependence is generated by the weights matrices M ¼ ðmijÞ and

W ¼ ðwijÞ, i; j ¼ 1; 2. . .;N, that have zero diagonal elements. The testing parame-

ters of interest are random effects (r2l0), serial correlation (q0 with jq0j\1), spatial

lag dependence (s0) and spatial error dependence (k0). The regression coefficient

vector b0 and innovation variance r2e0 are the nuisance parameters.

Let yt ¼ ðy1t; y2t; . . .; yNtÞ0, Xt ¼ ðX1t;X2t; . . .;XNtÞ0 and ut ¼ ðu1t; u2t; . . .; uNtÞ0. Inmatrix form, the equation in (3.1a) can be written compactly as

y ¼ s0ðIT �MÞyþ Xb0 þ u; ð3:2Þ

where y ¼ ðy01; y02; . . .; y0TÞ0is the NT � 1 vector of dependent variable, X ¼

ðX01;X

02; . . .;X

0TÞ

0is the NT � k matrix of exogenous variables, u ¼ ðu01; u02; . . .; u0TÞ

0

is the NT � 1 vector of regression disturbance terms, IT denotes the T � T identity

matrix, and � denotes the Kronecker product operator. Let l ¼ ðl1;l2; . . .; lNÞ0and

vt ¼ ðv1t; v2t; . . .; vNtÞ0. Then, using (3.1b) and (3.1c), the disturbance term can be

expressed as

u ¼ ðlT � INÞlþ IT � B�1ðk0Þ� �

v; ð3:3Þ

where Bðk0Þ ¼ ðIN � k0WÞ, lT is the T � 1 vector of ones and v ¼ ðv01; v02; . . .; v0TÞ0is

the NT � 1 vector of disturbance terms.

We use h0 ¼ ðb00; r2e0; r2l0; q0; k0; s0Þ0to denote the true parameter vector, and use

h ¼ ðb0; r2e ; r2l; q; k; sÞ0to denote any arbitrary value in the parameter space. Let

#0 ¼ k0; q0; r2l0; r

2e0

� �0be a sub-vector of h0. Under the assumption that l and

�t ¼ ð�1t; �2t; . . .; �ntÞ0 are independent for t ¼ 1; 2; . . .; T , the variance-covariance

matrix of u is given by

2 The regressors vector Xit can include the observations on (i) location and time varying variables, (ii)

time invariant, but location varying variables and (iii) location invariant, but time varying variables.

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Xð#0Þ ¼ r2l0ðJT � INÞ þ r2e0 Vq0 � ðB0ðk0ÞBðk0ÞÞ�1� �

; ð3:4Þ

where JT ¼ lT l0T is the T � T matrix of ones, and Vq0 is the familiar T � T variance-

covariance matrix for AR (1) process in (3.1d) defined by

Vq0 ¼1

1� q20

1 q0 q20 . . . qT�10

q0 1 q0 . . . qT�20

..

. ... ..

. . .. ..

.

qT�10 qT�2

0 qT�30 . . . 1

0BBBB@

1CCCCA:

The log-likelihood function of our model can be written as:

LðhÞ ¼ �NT

2ln 2p� 1

2ln jXð#Þj þ T ln jAðsÞj

� 1

2ðIT � AðsÞÞy� Xbð Þ0X�1ð#Þ ðIT � AðsÞÞy� Xbð Þ;

ð3:5Þ

where AðsÞ ¼ ðIN � sMÞ. Using Magnus (1982, Lemma 2.2) and some well-known

properties of Vq (see Online Appendix), it can be shown that

1

2ln jXð#Þj ¼ �N

2lnð1� q2Þ þ 1

2ln d2ð1� qÞ2/IN þ B0ðkÞBðkÞð Þ�1��

��

þ NT

2ln r2e � ðT � 1Þ ln jBðkÞj;

where d2 ¼ a2 þ ðT � 1Þ, a ¼ffiffiffiffiffiffiffi1þq1�q

qand / ¼ r2l

r2e. Substituting 1

2ln jXð#Þj into LðhÞ

yields

LðhÞ ¼ �NT

2ln 2pþN

2lnð1� q2Þ � 1

2ln d2ð1� qÞ2/IN þ B0ðkÞBðkÞð Þ�1��

��NT

2lnr2e

þ ðT � 1Þ ln jBðkÞj þ T ln jAðsÞj � 1

2u0ðhÞX�1ð#ÞuðhÞ;

ð3:6Þ

where uðhÞ ¼ ðIT � AðsÞÞy� Xb. In (3.6), X�1ð#Þ can be expressed as (see Magnus

(1982, Lemma 2.2))

X�1ð#Þ ¼ 1

d2ð1� qÞ2V�1q JTV

�1q � D�1 þ V�1

q � 1

d2ð1� qÞ2V�1q JTV

�1q

!

� 1

r2eB0ðkÞBðkÞ

� ;

where D ¼ r2e B0ðkÞBðkÞð Þ�1þr2ld2ð1� qÞ2IN

� �. The ML estimator (MLE) bh of h0

is obtained from the maximization of (3.6). We assume the following assumptions

for the asymptotic properties of bh.

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Assumption 1 (i) The weight matrices W and M have non-stochastic elements with

zero diagonal elements, and they are uniformly bounded in both row and column

sums in absolute value. (ii) BðkÞ ¼ IN � kWð Þ and AðsÞ ¼ IN � sMð Þ are invertiblefor all k 2 Dk and s 2 Ds, where Dk and Ds are compact parameter spaces.

Furthermore, k0 and s0 are, respectively, in the interiors of Dk and Ds. (iii) B�1ðkÞ ¼

IN � kWð Þ�1and A�1ðsÞ ¼ IN � sMð Þ�1

are uniformly bounded in both row and

column sums in absolute value for all k 2 Dk and s 2 Ds.3

Assumption 2 (i) The disturbances eits, i ¼ 1; . . .;N and t ¼ 2; . . .; T , are i.i.d.

normal across i and t with mean zero and variance r2e0. Furthermore, they are

independent with v0 ¼ v10; . . .; vN0ð Þ0 �N 0;r2e01�q2

0

IN

h i. (ii) the random effects li’s

are i.i.d. normal across i with mean zero and variance r2l0, and independent with v0,

(iii) li and vjt are mutually independent for all i, j and t.

Assumption 3 We assume that N is large and T is finite.

Assumption 4 The elements of the NT � k matrix X are non-stochastic and

bounded uniformly in N and T. Furthermore, the limit of 1NT X

0X�1ð#0ÞX exists and

is nonsingular under the asymptotic setting given in Assumption 3.

Assumption 5 (Identification) Consider the following cases: Case 1.

(i) limN!11NT Hð#Þ is nonsingular for a given value of #, where

Hð#Þ ¼ X;Gðs0ÞXb0ð Þ0X�1ð#Þ X;Gðs0ÞXb0ð Þ, Gðs0Þ ¼ IT � Gðs0Þ and Gðs0Þ ¼MA�1ðs0Þ, and (ii) limN!1 � 1

NT ln jXð#Þj � 1NT tr X�1ð#ÞXð#0Þ

� �þ

�1NT ln jXð#0Þj þ

1Þ 6¼ 0 for # 6¼ #0. Case 2. limN!11NT ln A

0�1ðs0ÞXð#0ÞA�1��

�ðs0Þj �

1NT ln A

0�1ðsÞXð#ÞA�1ðsÞ�� T ðs; #Þ þ 1Þ 6¼ 0 for ðs; #Þ 6¼ ðs0; #0Þ, where

T ðs; #Þ ¼ 1NT tr AðsÞA�1ðs0Þ

� �0X�1ð#Þ AðsÞA�1ðs0Þ

� �Xð#0Þ

� �, Aðs0Þ ¼

IT � Aðs0Þð Þ and AðsÞ ¼ IT � AðsÞð Þ. Then, either Case 1(i) and Case 1(ii) hold,or Case 2 holds if Case 1(i) fails.

Assumption 6 The limit of the information matrix is non-singular under the

asymptotic setting given in Assumption 3.

Assumption 1 rules out the self-influence by requiring that the weight matrices

have zero diagonal elements. The uniform boundedness in row and column sums in

absolute value conditions in Assumption 1 are suggested in the literature to limit the

spatial correlation to a manageable degree (Kelejian and Prucha 2001, 2010).

Assumption 1 also ensures that the valid reduced form of the model exists by

requiring that AðsÞ and BðkÞ are non-singular matrices. The compactness of

parameter spaces Ds and Dk is imposed in the literature for theoretical analysis on

nonlinear functions. It is common in the literature to assume that v0 ¼

3 A sequence of n� n matrix fAng is uniformly bounded in row sum in absolute value if

supn 1 kAnk1\1, where k � k1 is the row sum norm. Similarly, fAng is uniformly bounded in

column sum in absolute value if supn 1 kAnk1\1, where k � k1 is the column sum norm.

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ðv10; . . .; vN0Þ0 �N 0;r2e01�q2

0

IN

h ias in Assumption 2. The object of central limit

theorem (CLT) for our model is a linear and quadratic function of ðl0; e0Þ0, wheree ¼ ðe01; e02; . . .; e0TÞ

0and et ¼ ðe1t; e2t; . . .; eNTÞ0. When these terms are simply i.i.d.,

the CLT in Kelejian and Prucha (2001, 2010) requires the existence of ð4þ gÞthmoments for l and e, where g[ 0. The asymptotic setting in Assumption 3 requires

that T is fixed and N ! 1. However, our results are also valid under the large T

case (for details, see the proof of Proposition 1 in Online Appendix).

Assumption 4 is standard and allows for the regressors to be non-stochastic and

uniformly bounded (Kapoor et al. 2007; Yu et al. 2008). If the regressors are

allowed to be stochastic and unbounded, appropriate moment conditions can be

imposed as in Aquaro et al. (2019). Assumption 5 gives the identification conditions

for our model under the large N case. The identification uniqueness of h0 requires

that either (a) Case 1 holds, or (b) Case 2 holds if Case 1(i) fails. When (a) holds,

the identification of b0 and s0 is ensured by Case 1(i) and that of #0 by Case 1(ii).On the other hand, when (b) holds, the identification of s0 and #0 is ensured by Case2. Once s0 is identified, the identification of b0 is possible even when

limN!11NT Hð#Þ is a singular matrix (for the details, see proof Proposition 1 given

in Online Appendix). Finally, Assumption 6 requires that the limit of the

information matrix is non-singular.

Under our stated assumptions, we establish the consistency and asymptotic

normality of bh as shown in the following proposition.

Proposition 1 Under Assumptions 1–6, as N ! 1, the MLE bh of h0 is consistentand has the asymptotic normal distribution, namely,

ffiffiffiffiffiffiffiNT

p bh � h0� �

!d N 0; J�1ðh0Þ �

; ð3:7Þ

where Jðh0Þ ¼ limN!1 E � 1NT

o2Lðh0Þohoh0

h i.

Proof See Online Appendix C. h

Remark 1 Our model can be turned into a spatial dynamic panel data model by

including a time lag and a spatial-time lag of the dependent variable. The spatial

dynamic models are subject to two well-known problems, namely (i) the initial-

value problem and (ii) the incidental parameter problem due to the presence of fixed

effects. Although, under the large N and large T setting, both problems can be

avoided as shown by Yu et al. (2008), the asymptotic distribution of score functions

is not centered around zero, i.e., the (Q)MLE has an asymptotic bias. Bera et al.

(2019b) systematically show how to develop score based tests in this setting. In the

case, where N is large but T is fixed, Su and Yang (2015) show that the incidental

parameter problem can still be avoided by a suitable transformation to wipe-out the

individual fixed effects from the model, but the initial-value problem will persist

and the asymptotic properties of (Q)MLE will depend on the assumption adopted

for the initial-values. See also Elhorst (2005, 2010a). Furthermore, under the large

N and small T case, the asymptotic variance of score functions can not be estimated

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easily by the traditional methods based on the sample-analogues, outer product of

gradients (OPG), or closed form expressions. Su and Yang (2015) overcome this

difficulty by suggesting a simple residual-based bootstrap method for inference.

Zhenlin (2018) suggests an alternative approach based on the M-estimation

framework for the large N and small T setting. This approach is free from the

specification of the distribution of the initial observations and is robust against

nonnormality of the errors. It is clear that the inclusion of a time lag and a spatial-

time lag of the dependent variable into our specification will introduce several

complications, and therefore we prefer to leave it for a future study.

4 Specification tests

4.1 Rao’s score test under parametric misspecification

Let us denote h ¼ ðc0;w0;/0Þ0, where c ¼ ðb0; r2eÞ0and w and / can be any

combination of remaining parameters, namely fr2l; q; k; sg. For simplicity, we

assume that w and / are, respectively, r � 1 and s� 1 vectors such that r þ s ¼ 4.

Let daðhÞ ¼ 1NT

oLðhÞoa ; a ¼ fc;w;/g and consider the following partition

JðhÞ ¼ E � 1

NT

o2LðhÞohoh0

� ¼

JcðhÞ JcwðhÞ Jc/ðhÞJwcðhÞ JwðhÞ Jw/ðhÞJ/cðhÞ J/wðhÞ J/ðhÞ

0B@

1CA: ð4:1Þ

At the true parameter vector, we use da ¼ daðh0Þ and J ¼ Jðh0Þ for the notational

simplicity. Consider the log-likelihood function Lðc;w;/Þ and suppose that a

researcher assumes H/0 : /0 ¼ /H and tests Hw

0 : w0 ¼ wH using the log-likelihood

function L1ðc;wÞ ¼ Lðc;w;/HÞ, where wH and /H are known quantities and most

of the time they will be zeros. The RS statistic for testing Hw0 in L1ðc;wÞ will be

denoted by RSw. Let ~h ¼ ð~c0;w0H;/0

HÞ0 be the MLE of h0 under H

w0 and H/

0 . If

L1ðc;wÞ were the true model, then it is well known that under Hw0 , we have

RSw ¼ NTd0wð~hÞJ�1w�cð~hÞdwð~hÞ!

dv2r ; ð4:2Þ

where Jw�cð~hÞ ¼ Jwð~hÞ � Jwcð~hÞJ�1c ð~hÞJcwð~hÞ. Consider the sequence of local alter-

native hypothesis HwA : w0 ¼ wH þ fffiffiffiffiffi

NTp , where f is a non-stochastic bounded vector.

It can be shown that under HwA , RSw!

dv2r ðk1Þ, where the non-centrality parameter is

given by

k1 ¼ f0Jw�cf: ð4:3Þ

Given this setting, i.e., under no misspecification, asymptotically the test will have

the correct size and locally be optimal. Now suppose that the true log-likelihood

function is L2ðc;/Þ ¼ Lðc;wH;/Þ so that the considered alternative L1ðc;wÞ is

123


(completely) misspecified. Using the local misspecification H/A : /0 ¼ /H þ dffiffiffiffiffi

NTp ,

where d is a bounded vector, Davidson and MacKinnon (1987) and Saikkonen

(1989) derived the asymptotic distribution of RSw under L2ðc;/Þ as RSw!dv2r ðk2Þ,

where the non-centrality parameter k2 is

k2 ¼ d0J/w�cJ�1w�cJw/�cd; ð4:4Þ

with Jw/�c ¼ Jw/ � JwcJ�1c Jc/. Owing to the presence of this non-centrality

parameter k2;RSw will reject the true null hypothesis Hw0 : w0 ¼ wH more often, i.e.,

the test will have excessive size. Here the crucial term is J/w�c which can be

interpreted as the partial covariance between the score vectors d/ and dw after

eliminating the linear effects of dc on d/ and dw. If Jw/�c ¼ 0, then the local presence

of /0 has no effect on RSw asymptotically.

We adjust RSw to overcome this problem of over-rejection, so that the resulting

test is valid under the local presence of /0. The modified RS statistic for testing

Hw0 : w0 ¼ wH takes the following form.

RSw ¼ NTd0

w ð~hÞ Jw�cð~hÞ � Jw/�cð~hÞJ�1/�cð~hÞJ0w/�cð~hÞ

� ��1

dwð~hÞ; ð4:5Þ

where dwð~hÞ ¼ dwð~hÞ � Jw/�cð~hÞJ�1/�cð~hÞd/ð~hÞ

� �is the adjusted score function and

J/�cð~hÞ ¼ J/ð~hÞ � J/cð~hÞJ�1c ð~hÞJc/ð~hÞ. This new test essentially adjusts the mean

and variance of the standard RS statistics RSw. In the following proposition, we give

asymptotic distribution of RSw along with the results summarized so far.

Proposition 2 Assume that Assumptions 1–6 hold. Then, as N ! 1, the followingresults hold.

1. Under HwA and H/

A , we have

RSw!dv2r n1ð Þ; ð4:6Þ

where n1 ¼ f0Jw�cfþ 2f0Jw/�cdþ d0J0w/�cJ�1w�cJw/�cd is the non-centrality

parameter.

2. Under Hw0 , it follows that

RSw!dv2r : ð4:7Þ

3. Under HwA , we have

RSw!dv2r n2ð Þ; ð4:8Þ

where n2 ¼ f0 Jw�c � Jw/�cJ�1/�cJ

0w/�c

� �f is the non-centrality parameter.

123


Proof See Online Appendix C. h

Proposition 2 shows that n1 ¼ k1 þ k2 þ 2f0Jw/�cd under HwA and H/

A , where k1 ¼f0Jw�cf and k2 ¼ d0J/w�cJ�1

w�cJw/�cd. The results in the last two parts are valid under

the presence or absence of local misspecification, since the asymptotic distribution

of RSw is unaffected by the local departure of /0 from /H. That is, these results hold

irrespective of whether H/0 or H/

A holds. Our adjusted test is asymptotically

equivalent to Neyman’s CðaÞ test and thus shares its optimal properties (Bera and

Yoon 1993). Three observations are worth noting regarding RSw. First, RSw requires

estimation only under the joint null, namely Hw0 and H/

0 . That means, as we will see

later, we can compute the adjusted tests from the OLS estimates. Given the full

specification of the model Lðc;w;/Þ, it is of course possible to derive RS test for

testing Hw0 after estimating /0 (and c0) by the MLE bh, which are generally referred

to as the conditional tests. However, the ML estimation of /0 could be difficult in

some instances. Second, when Jw/�c ¼ 0, which is a simple condition to check,

RSw ¼ RSw and thus RSw is an asymptotically valid test in the local presence of /0.

Finally, let RSw/ denote the joint RS test statistic for testing hypothesis of the form

H0 : w0 ¼ wH and /0 ¼ /H using the alternative model Lðc;w;/Þ, then it can be

shown that (Bera et al. 2009)

RSw/ ¼ RSw þ RS/ ¼ RS/ þ RSw; ð4:9Þ

where RS/ and RS/ are, respectively, the counterparts of RSw and RSw for testing

H/0 : /0 ¼ /H. This is a very useful identity since it implies that a joint RS test for

two parameter vectors w and / can be decomposed into sum of two orthogonal

components: (i) the adjusted statistic for one parameter vector and (ii) the (unad-

justed) marginal test statistic for the other. Many econometrics softwares provide

the marginal test statistics RSw and RS/, and sometime the joint test RSw/, therefore,

the adjusted versions can be obtained effortlessly.

Significance of RSw/ indicates some form of misspecification in the basic model

with parameter vector c only. However, the correct source(s) of departure can be

identified only by using the adjusted statistics RSw and RS/ not the marginal ones

(RSw and RS/). This testing strategy is close to the idea of Hillier (1991) in the sense

that it partitions the overall rejection region to obtain evidence about the specific

direction(s) in which the basic model needs revision. And it achieves that without

estimating any of the nuisance parameters.

4.2 Score functions and information matrix

We are interested in testing H0 : w0 ¼ 0 in the possible presence of the parameter

vector /0. In the context of our earlier notation, h ¼ ðc0;w0;/0Þ0, c ¼ ðb0; r2eÞ0and w

and / could be any combinations of the parameters under test, namely fr2l; q; k; sg.The main advantage of using RS test principal is that we need estimation of h0 only

under the joint null Ha0 : r2l0 ¼ q0 ¼ k0 ¼ s0 ¼ 0. Let ~h ¼ ð~b0; ~r2e ; 0; 0; 0; 0Þ

0be the

123


MLE under Ha0 . From (3.5), we have ~b ¼ ðX0XÞ�1X0Y and ~r2e ¼ ~u0 ~u=NT , where

~u ¼ y� X~b. For simplicity, we assume that M ¼ W . This is often realistic in

practice, since there may be good reasons to expect the structure of spatial

dependence to be the same for the dependent variable y and the innovation term �.On the basis of the derivations given in Online Appendix B, the score functions

evaluated at ~h are (we omit the normalization by NT for simplicity):

dbð~hÞ ¼ 0k�1; dr2e ð~hÞ ¼ 0; dr2lð~hÞ ¼NT

2~r2e

~u0ðJT � INÞ~u~u0 ~u

� 1

� ;

dqð~hÞ ¼NT

2

~u0ðFqð~hÞ � INÞ~u~u0 ~u

!; dkð~hÞ ¼

NT

2

~u0 IT � ðW þW 0Þð Þ~u~u0 ~u

� ;

dsð~hÞ ¼~u0ðIT �WÞy

~r2e;

ð4:10Þ

where Fqð~hÞ ¼ oVq=oq� �

j~h is the T � T bidiagonal matrix with bidiagonal elements

all equal to one. The information matrix J evaluated at ~h is (we omit the normal-

ization by NT for simplicity, see Online Appendix B)

Jð~hÞ ¼

X0X

~r2e0 0 0 0

X0ðIT �WÞX~b~r2e

0NT

2~r4e

NT

2~r4e0 0 0

0NT

2~r4e

NT2

2~r4e

NðT � 1Þ~r2e

0 0

0 0NðT � 1Þ

~r2eNðT � 1Þ 0 0

0 0 0 0 TtrðW2 þWW 0Þ T trðW2 þWW 0Þ~b0X0ðIT �W 0ÞX

~r2e0 0 0 TtrðW2 þWW 0Þ H

0

BBBBBBBBBBBBBBBBBBBB@

1

CCCCCCCCCCCCCCCCCCCCA

;

ð4:11Þ

where H ¼ T trðW2 þWW 0Þ þ ~b0X0ðIT�W 0ÞðIT�WÞX ~b~r2e

. Apart from the RS statistic for full

joint null hypothesis Ha0 , we propose four (modified) test statistics for the following

hypotheses:

1. Hb0 : r2l0 ¼ 0 in the presence of q0, k0, s0;

2. Hc0 : q0 ¼ 0 in the presence of r2l0, k0, s0;

3. Hd0 : k0 ¼ 0 in the presence of r2l0, q0, s0;

4. He0 : s0 ¼ 0 in the presence of r2l0, q0, k0.

These four tests will guide us to identify the correct source(s) of departure(s) from

Ha0 : r2l0 ¼ q0 ¼ k0 ¼ s0 ¼ 0 when it is rejected. One can test various combinations

by testing two/three parameters at a time under the null and compute additional ten

test statistic (as is done sometimes in practice). However, we would argue that is not

necessary. Also keeping the total number of tests to a minimum is beneficial to

123


avoid the pre-testing problem since in practice, researchers reformulate their model

based on test outcomes. Given the full specification of the model in (3.1a)–(3.1d), it

is of course possible to derive conditional RS and LR tests, for say, Hb0 : r2l0 ¼ 0 in

the presence of q0, k0, s0, as advocated in Baltagi et al. (2003), Baltagi et al. (2007)

and Baltagi and Liu (2008). However, that requires ML estimation of ðq0; k0; s0Þ0(and also of r2l0 for the LR test). The focus of our strategy is to carry out the

specification test for our general model with minimum estimation. As we will see

later from our Monte Carlo results in Sect. 6, we lose very little in terms of finite

sample size and power. Though RSr2ldoes not require explicit estimation of

/0 ¼ ðq0; k0; s0Þ0, effect of these parameters have been taken into account through

the use of the effective score function. Of course, given the current computing

power, it is not that difficult to estimate a complex model. However, it could be at

times hard to ensure the stability of many parameter estimates. Also, theoretically,

the stationarity regions of the parameter space have not been fully worked out as

discussed in Elhorst (2010b).

4.3 Adjusted RS tests

We now discuss the test statistics for each of the above hypotheses by using

Proposition 2 (all derivations details are given in Online Appendix D).

4.3.1 Testing Hb0:r

2l0 = 0 in the presence of q0, k0, s0

Here we are testing the significance of random location/individual effect in the

presence of time series autocorrelation of errors, spatial error dependence and

spatial lag dependence. In terms of our notation, here w ¼ r2l, / ¼ ðq; k; sÞ0 andc ¼ ðb0; r2eÞ

0, and we have

Jw/�c ¼ Jr2lq; 0; 0� �

¼ NðT � 1Þ=r2e0; 0; 0� �

;

which implies that the unadjusted RS is not a valid test under the local presence of

/0. However, note that only the partial covariance between dr2l and dq is nonzero,

while it is zero for dr2l and dk; dr2l and ds. This fact gets reflected in the unadjusted

and adjusted version of the test statistic for Hb0 :

RSr2l ¼NTA2

1

2ðT � 1Þ ;ð4:12aÞ

RSr2l ¼NT2ðA1 � B1Þ2

2ðT � 1ÞðT � 2Þ ;ð4:12bÞ

where A1 ¼ ~u0 JT�INð Þ ~u~u0 ~u � 1 and B1 ¼

~u0 Fqð~hÞ�INð Þ ~u~u0 ~u . The robust test RSr2l

adjusts the

score dr2lð~hÞ such that the effective (adjusted) score dr2lð~hÞ is the part of dr2lð~hÞ that

123


is ‘‘orthogonal’’ to dqð~hÞ. Loosely, we can write dHr2lð~hÞ ¼ dr2lð~hÞ � E dr2lð~hÞjdqð~hÞ

h i.

For the other nuisance parameters, k (spatial error lag) and s (spatial dependence

lag), such adjustments are not needed since they do not have any asymptotic effect

on r2l as far as the testing is concerned. Similar interpretation applies to the variance

of the adjusted score, which essentially reflects the adjustment needed in variance

part for changing the raw score to effective score. Thus inference regarding r2l is

affected only by the presence of q0 and is independent of the spatial aspects of the

model. This separation between ‘‘time’’ and ‘‘spatial’’ aspects of the panel spatial

model is quite interesting, and we consider it as a plus that our adjusted test take

account of such information implied by model.

Using (4.6) and (4.8) in Proposition 2, we determine the non-centrality

parameters of RSr2l and RSr2lin the following corollary.4

Corollary 1 Assume that Assumptions 1–6 hold. Then, as N ! 1, the followingresults hold.

(a) Under HwA and H/

A , we have RSr2l�!dv21ðn1Þ, where n1 ¼ k1 þ k2 þ 2f0Jw/�cd

with k1 ¼ r4l0ðT � 1Þ=2r4e0, k2 ¼ 2q20ðT � 1Þ=T2 and

2f0Jw/�cd ¼ 2r2l0q0ðT � 1Þ=Tr2e0.

(b) Under HwA , we have RSr2l

�!d v21ðn2Þ, where n2 ¼ r4l0ðT � 1ÞðT � 2Þ=2r4e0T .

Proof See Online Appendix E. h

Note that the non-centrality parameters in Corollary 1 are the same as with those

derived in Bera et al. (2001) for a non-spatial one-way error component model.

Corollary 1 gives n1 ¼ k2 ¼ 2q20ðT � 1Þ=T2 under Hw0 and H/

A , showing that RSr2l

will reject the true Hw0 too often when q0 6¼ 0. On the other hand, RSr2l

has no such

problem as its null distribution is a central chi-squared distribution irrespective of

whether H/0 or H/

A holds. Under HwA and H/

A , Corollary 1 shows that the change in

the non-centrality parameter of RSr2l due to nonzero q0 is given by

n1 � k1 ¼ k2 þ 2f0Jw/�cd ¼ 2ðT � 1ÞT

q20T

þr2l0q0r2e0

!: ð4:13Þ

This result indicates that the presence of autocorrelation (as is usually the case for

economic data) can increase or decrease the power of RSr2l , depending on the size of

q20=T þ r2l0q0=r2e0

� �. Under Hw

A and H/0 , the difference between the non-centrality

parameters of RSr2l and RSr2lis given by

4 In reporting the non-centrality parameters, we use w and / for f and d, respectively.

123


n1 � n2 ¼ k1 � n2 ¼r4l0ðT � 1Þ

r4e0T;

which is the cost of applying RSr2lwhen there is no misspecification, i.e., when q0 is

indeed zero.

4.3.2 Testing Hc0:q0 = 0 in the presence of r2l0, k0, s0

Here, we test the significance of time-series autocorrelation in presence of random

effect, spatial lag and spatial error dependence effects, i.e., w ¼ q and

/ ¼ ðr2l; k; sÞ0. It can be shown that

Jw/�c ¼ Jqr2l ; 0; 0� �

¼ NðT � 1Þ=r2e0; 0; 0� �

:

Again this expression can be given similar interpretation as above, i.e., the inference

on q0 will be affected only by the presence of random effect, not by the presence of

spatial dependence. The unadjusted and adjusted test statistics for this case are:

RSq ¼ NT2B21

4ðT � 1Þ ;ð4:14aÞ

RSq ¼ NT2 B1 � 2A1=Tð Þ2

4ðT � 1Þð1� 2=TÞ :ð4:14bÞ

In the following corollary, we provide the non-centrality parameters of RSq and

RSq.



A , we have RSq !dv21ðn1Þ, where n1 ¼ k1 þ k2 þ 2f0Jw/�cd

with k1 ¼ q20ðT � 1Þ=T , k2 ¼ r4l0ðT � 1Þ=r4e0T and 2f0Jw/�cd

¼ 2r2l0q0ðT � 1Þ=Tr2e0.(b) Under Hw

A , it follows that RSq !d v21ðn2Þ, where n2 ¼ q20ðT � 1ÞðT � 2Þ=T2.


Since the inference on q0 is not affected by the spatial aspects of our model, the

non-centrality parameters in Corollary 2 are the same as with those derived in Bera

et al. (2001) for a non-spatial one-way error component model. Under Hw0 and H/

A ,

we have n1 ¼ k2 ¼ r4l0ðT � 1Þ=r4e0T , indicating that RSq will reject Hw0 too often.

Under HwA and H/

A , Corollary 2 shows that the change in the non-centrality

parameter of RSr2l due to the presence of random effects is

123


n1 � k1 ¼ k2 þ 2f0Jw/�cd ¼ðT � 1Þr2l0

Tr2e0

r2l0r2e0

þ 2q0

!; ð4:15Þ

which indicates that the power of RSq can increase or decrease depending on the

size of r2l0=r2e0 þ 2q0

� �. Finally, the cost of applying RSq in the absence of random

effects is given by

n1 � n2 ¼ k1 � n2 ¼ 2q20ðT � 1Þ=T2:

Thus, for large values of T, this cost will be small.

4.3.3 Testing Hd0 :k0 = 0 in the presence of r2l0, q0, s0

In terms of our notation Proposition 2, we have w ¼ k and / ¼ ðr2l; q; sÞ0and

Jw/�c ¼ 0; 0; Jksð Þ;

where Jks ¼ T tr W2 þWW 0ð Þ. The test statistics are

RSk ¼1

Ttr W2 þWW 0ð Þ

�NT ~u0

�IT � ðW þW 0Þ

�~u

2~u0 ~u

2

; ð4:16aÞ

RSk ¼12~r2e

~u0 IT � ðW þW 0Þð Þ~u� 2T tr W2 þWW 0ð ÞJ�1s�bð~hÞ~u0ðIT �WÞy

� �� 2

T tr W2 þWW 0ð Þ 1� T tr W2 þWW 0ð ÞJ�1s�bð~hÞ

� � ;

ð4:16bÞ

where Js�bð~hÞ ¼ H � ~b0X0ðIT �W 0ÞXðX0XÞ�1X0ðIT �WÞX~b�~r2e

� �.

The non-centrality parameters for RSk and RSk are stated in the following

corollary.



A , we have RSk�!dv21ðn1Þ, where n1 ¼ k1 þ k2 þ 2f0Jw/�cd

with k1 ¼ s20Jk=NT , k2 ¼ s0Jks=NT and 2f0Jw/�cd ¼ 2k0s0Jks=NT .

(b) Under HwA , we have RSk�!

dv21ðn2Þ, where n2 ¼ k20Jks 1� JksJ

�1s�b

� �=NT .


Under Hw0 and H/

A , Corollary 3 gives n1 ¼ k2 ¼ s0Jks=NT , indicating that RSk

will reject Hw0 too often. Under Hw

A and H/A , Corollary 3 shows that the change in the

non-centrality parameter of RSk due to the presence of s0 is given by

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n1 � k1 ¼ k2 þ 2f0Jw/�cd ¼ s0Jksð1þ 2k0Þ=NT : ð4:17Þ

Finally, the cost of applying RSk when s0 is indeed zero, i.e., when HwA and H/

0 hold,

is given by

n1 � n2 ¼ k1 � n2 ¼ k20J2ksJ

�1s�b=NT :

Thus, the cost will be lower for large sample size as we will notice in our simulation

studies in Sect. 6.

4.3.4 Testing He0:s0 = 0 in the presence of r2l0, q0, k0

In this case, we set w ¼ s and / ¼ ðr2l; q; kÞ0and find that

Jw/�c ¼ 0; 0; Jskð Þ;

where Jsk ¼ T tr W2 þWW 0ð Þ. The test statistics are

RSs ¼~u0ðIT �WÞyð Þ2

~r4eJs�bð~hÞ; ð4:18aÞ

RSs ¼12~r2e

2~u0ðIT �WÞy� ~u0 IT � ðW þW 0Þð Þ~uð Þ� �2

Js�bð~hÞ � T tr W2 þWW 0ð Þ: ð4:18bÞ

Using Proposition 2, we determine the non-centrality parameters of RSs and RSs inthe following corollary.



A , we have RSs�!dv21ðn1Þ, where n1 ¼ k1 þ k2 þ 2f0Jw/�cd

with k1 ¼ s20Js�b=NT , k2 ¼ k20J2skJ

�1s�b=NT and 2f0Jw/�cd ¼ 2k0s0Jsk=NT .

(b) Under HwA , we have RSs�!

dv21ðn2Þ, where n2 ¼ s20ðJs�b � JskÞ=NT .


Under Hw0 and H/

A , Corollary 4 gives n1 ¼ k2 ¼ k20J2skJ

�1s�b=NT . Under H

wA and H/

A ,

Corollary 4 shows that the change in the non-centrality parameter of RSs due to the

presence of k0 is given by

n1 � k1 ¼ k2 þ 2f0Jw/�cd ¼ k0Jsk k0JskJ�1s�b þ 2s0

� �=NT : ð4:19Þ

Finally, the cost of applying RSs when k0 is indeed zero, i.e., when HwA and H/

0 hold,

is given by

123


n1 � n2 ¼ s20Jsk=NT :

4.3.5 Relationships among tests

Recall that when Jw/�c ¼ 0 holds, we show that RSw ¼ RSw and thus RSw is an

asymptotically valid test in the local presence of /0. In the following corollary, we

summarize the orthogonality conditions that hold in the context of our model.

Corollary 5 Let h0 ¼ ðc0; 0; 0; 0; 0Þ0, where c0 ¼ ðb00; r2e0Þ0. Then, it follows that (i)

JqðksÞ�r2lc ¼ 0, (ii) Jr2lðksÞ�qc ¼ 0, (iii) Jkðr2lqÞ�sc ¼ 0, (iv) Jsðr2lqÞ�kc ¼ 0 and (v)

JðksÞðr2lqÞ�c ¼ 0.


The orthogonality conditions JqðksÞ�r2lc ¼ 0 and Jr2lðksÞ�qc ¼ 0 suggest that RSq and

RSr2l are valid tests in the local presence of k0 and s0. Similarly, it follows from the

conditions Jkðr2lqÞ�sc ¼ 0 and Jsðr2lqÞ�kc ¼ 0 that RSk and RSs are valid tests in the

local presence of r2l0 and q0. The orthogonality between non-spatial features of the

model ðr2l and q) and its spatial counterpart (k and s) in Corollary 5 suggests that

the joint RS statistic Rr2lqks can be decomposed as RSr2lqks ¼ RSr2lq þ RSks. In the

following corollary, we formally state this result and show how Rr2lqks can be

expressed in terms of marginal RS statistics.

Corollary 6 Assume that Assumptions 1–6 hold. Let w ¼ ðr2l; q; k; sÞ0 and

c ¼ ðb0; r2eÞ0. Then, under Ha

0 , it follows that

Rr2lqks ¼ d0wð~hÞJ�1w�cð~hÞdwð~hÞ�!

dv24;

where dwð~hÞ ¼ dr2lð~hÞ; dqð~hÞ; dkð~hÞ; dsð~hÞ� �0

and

Jw�cð~hÞ ¼

NTðT � 1Þ2~r4e

NðT � 1Þ~r2e

0 0

NðT � 1Þ~r2e

NðT � 1Þ 0 0

0 0 TtrðW2 þWW 0Þ TtrðW2 þWW 0Þ0 0 TtrðW2 þWW 0Þ Js�bð~hÞ

0

BBBBBBBB@

1

CCCCCCCCA

;

with Js�bð~hÞ ¼ H � ~b0X0ðIT �W 0ÞXðX0XÞ�1X0ðIT �WÞX~b�~r2e . The RSr2lqks statistic

can be decomposed as

123


RSr2lqks ¼ RSr2lq þ RSks ¼ RSr2l þ RSq þ RSk þ RSs ¼ RSr2l þ RSq þ RSk þ RSs :

ð4:20Þ


As expected, Corollary 6 shows that the omnibus test statistic RSr2lqks is not the

sum of four marginal RS statistics, i.e., RSr2lqks 6¼ RSr2l þ RSq þ RSk þ RSs. This

result supports our finding that the unadjusted RS statistics over-reject the respective

nulls as they fail to take into account of the effect of the relevant interaction effects

within the spatial and non-spatial parameters. From the decomposition given in

Corollary 6, we can trivially obtain the adjusted RS tests from their unadjusted

counterparts as shown in the following corollary.

Corollary 7 The following results hold for the the adjusted RS tests.

RSr2l ¼ RSr2lq � RSq; ð4:21Þ

RSq ¼ RSr2lq � RSr2l ; ð4:22Þ

RSk ¼ RSks � RSs; ð4:23Þ

RSs ¼ RSks � RSk; ð4:24Þ

where

Rr2lq¼

dr2lð~hÞ

dqð~hÞ

!0NTðT � 1Þ

2~r4e

NðT � 1Þ~r2e

NðT � 1Þ~r2e

NðT � 1Þ

0BBB@

1CCCA

�1

dr2lð~hÞ

dqð~hÞ

!;

and

Rks ¼dkð~hÞdsð~hÞ

!0T trðW2 þWW 0Þ T trðW2 þWW 0ÞT trðW2 þWW 0Þ Js�bð~hÞ

!�1

dkð~hÞdsð~hÞ

!:


Corollary 7 provides a substantial computational simplicity for practitioners. One

can easily obtain the joint RS (two directional) and marginal RS (one directional)

for the parameters using any popular statistical package like STATA, R, Matlab and

Python based on the OLS residuals, and then obtain the adjusted test statistics as

above. Thus, our methodology is implementable without any computational burden,

unlike the LR and conditional RS (LM) tests.

123


Remark 2 Our suggested tests are derived under the assumption that the random

effects and the disturbance terms are normally distributed. If we assume that these

terms are simply i.i.d with the additional properties, Ejvitj4þj1\1 and

Ejlij4þj2\1 for some j1 [ 0 and j2 [ 0, then the conclusion in Proposition 1

takes the following form

ffiffiffiffiffiffiffiNT

p bh � h0� �

!d N 0; J�1ðh0Þ Jðh0Þ þ Kðh0Þð ÞJ�1ðh0Þ �

; ð4:25Þ

where Jðh0Þ þ Kðh0Þ ¼ limN!1 E 1NT

oLðh0Þoh

oLðh0Þoh0

h i, and Kðh0Þ is related to the third

and fourth moments of the random effects and the disturbance terms.5 The presence

of Kðh0Þ affects our test statistics. In a recent study, Bera et al. (2019a) show how to

derive the robust test statistic in the QML setting. Their analysis shows that though

the adjusted score function is not affected in the presence of Kðh0Þ, its asymptotic

variance depends on Kðh0Þ. Therefore, the effect of nonnormality on the finite

sample properties of our test statistics remains to be studied in a future study.

5 An empirical illustration

We now present an empirical application that illustrates the usefulness of our proposed

tests. The data consist of a sample of 91 countries over the period 1961–1995. These

countries are those from the Mankiw et al. (1992) non-oil sample, for which Heston

et al. (2002) Penn World Table (PWT version 6.1) provides data. We use a slight

variation of Ertur and Koch (2007)’s growth model that explicitly takes account of

technological interdependence among countries and examines the impact of neighbor-

hood effect. The magnitude of physical capital externalities at steady state, which is not

usually identified in the literature, is estimated using a spatially augmented Solow

model. Our aim is to illustrate how a practitioner, after estimating the simplest model,

would proceed to identify the spatial and non-spatial structures and reformulate the

model accordingly. We consider the following estimation equation:

lnYitLit

� ¼ b0 þ b1 ln sit þ b2 lnðnit þ gþ dÞ þ s

XN

j6¼i

wij lnYjtLjt

�

þXN

j6¼i

wijðb3 ln sjt þ b4 lnðnjt þ gþ dÞÞ þ uit;

uit ¼ li þ �it; �it ¼ kXN

j 6¼i

wij�jt þ vit; vit ¼ qvit�1 þ eit;

where Y is real GDP, L is the number of workers, s is the saving rate, and n is the

average growth of the working-age population (ages 15–64), d is the depreciation

rate of physical capital and g is the balanced growth rate. We set ðdþ gÞ ¼ 0:05 as

5 A closed form expression for Kðh0Þ is given in Lee and Yu (2012) for a model that nests our

specification.

123


is common in the literature. The (i, j)th element wij of W is based on geographical

distance, as in Ertur and Koch (2007).

We estimate the model by OLS under our joint null hypothesis, i.e., when all the

four effects are absent, and then compute the following test statistics: (i) the joint

test for all four departures, i.e. random effect, serial correlation, spatial error lag and

spatial lag, (RSr2lqks), (ii) the joint test for random effect and serial correlation

ðRSr2lq), (iii) the joint test for spatial error lag and lag dependence (RSks), (iv) the

Breusch–Pagan test for random effects (RSr2l ), (v) the proposed modified version

(RSr2l), (vi) the RS test of serial correlation (RSq), (vii) the corresponding modified

version (RSq), (viii) the RS test of spatial error dependence (RSk), (ix) the proposed

modified version (RSk) (x) the RS test of spatial lag dependence (RSs), and (xi)

lastly the modified version (RSs). To identify specific departure(s) there is no need

to consider any other combination of tests due to the asymptotic independence

discussed earlier. Here we are reporting the unadjusted RS tests, though they are

mainly for comparison purpose and are not necessary to carryout model

specification search in practice.

The test statistics are presented in Tables 1 and 2. All of the test statistics are

computed individually, and we verified the equalities in Eqs. (4.21)–(4.24). The

omnibus statistic ðRSr2lqks ¼ 220:02Þ rejects the joint null when compared to v24critical value at any level. Later in our Monte Carlo case study we will demonstrate

the good finite sample size of RSr2lqks. More specifically, its estimated and nominal

sizes are close for various settings of (N, T) values. From Table 1, RSr2lq ¼ 189:45

can be viewed as a measure of non-spatial departures for r2l and q, and similarly

RSks ¼ 30:57 is for spatial parameters, k and s. Both statistics are highly significant

after comparing them to v22 critical points. These joint tests are, however, not

informative about the specific direction(s) of the misspecification(s). All the

unadjusted statistics RSr2l , RSq, RSs and RSk strongly reject the respective null

hypothesis. If an investigator takes these rejections at their face values, then s/he

would attempt to incorporate all these four parameters into the final model.

However, as we pointed out these one-directional tests are not valid in the possible

presence of other effects. Significance of each parameter can only be evaluated

correctly by considering our modified tests. Three of the modified versions RSr2l,

Table 1 Joint testsRSr2lqks RSr2lq RSks

220.02 189.45 30.57

Table 2 Robust and non-robust

testsRSr2l RSr2l

RSq RSq RSk RSk RSs RSs

183.03 157.14 32.31 6.36 26.01 0.10 30.46 4.55

123


RSq and RSk still reject the respective null at 1% significance level, when compared

to v21 critical values, though it is interesting to see how the values of the statistics

have reduced after modification. A somewhat striking result is that the value of RSkis 0.10 in contrast to that of RSk ¼ 26:01. From our analytical results in the previous

section, it is clear that 26.01 is not only for the spatial error dependence but also

reflects the presence of lag dependence which seems to be much stronger for this

data set. Thus, the misspecification of the basic model can be thought to come from

the presence of random effects, serial correlation and spatial lag dependence (rather

than spatial error dependence) of the real income of the countries.

This empirical exercise seems to illustrate clearly the main points of the paper:

the proposed modified versions of RS tests are more informative than the

unmodified counterparts. It is worth noting a few observations from our analytical

and the empirical results. Since RSr2lqks ¼ RSr2lq þ RSks, the joint test for serial

correlation and random effect is independent of the joint test for spatial lag and

spatial error dependence. However, further additivity fails, as we note: RSr2lq 6¼RSr2l þ RSq and RSks 6¼ RSk þ RSs. This is due to the non-zero interaction effects

between parameters, and thus the unadjusted statistics are contaminated by the

presence of other parameters. We have

RSq þ RSr2l � RSr2lq ¼ RSr2l � RSr2l ¼ RSq � RSq ¼ 25:95;

RSk þ RSs � RSks ¼ RSk � RSk ¼ RSs � RSs ¼ 25:91:

It is important to emphasize again that the implementation of the modified tests is

based solely on OLS residuals and parameter estimates. Some currently available

test strategies relies on ML estimation of the general spatial panel model with all the

parameters, and then carrying out LR or conditional RS tests individually or jointly.

However, we propose asymptotically equivalent tests without estimating the com-

plex model at all. In the next section we demonstrate that though our suggested tests

are theoretically valid only for large samples and local misspecification, they per-

form quite well in finite samples and also for not-so-local departures. We also show

that a very little is lost in terms of size and power in using our simple tests compared

to the full-fledged (or partial-fledged, i.e., for conditional tests) computationally

demanding tests.

6 Monte Carlo results

To facilitate comparisons with existing results we follow a structure close to Baltagi

et al. (2007) and Baltagi and Liu (2008). The data were generated using the model:

yit ¼ a0 þ s0XN

j¼1

wijyjt þ Xitb0 þ uit; uit ¼ li þ �it; ð6:1Þ

123


�it ¼ k0XN

j¼1

wij�jt þ vit; vit ¼ q0vit�1 þ eit: ð6:2Þ

We set a0 ¼ 5 and b0 ¼ 2. The independent variable Xit is generated using:

Xit ¼ 0:1t þ 0:5Xit�1 þ uit; ð6:3Þ

where uit �Uniform½�0:5; 0:5� and Xi0 ¼ 5þ 10ui0. For the weight matrix W, we

consider the rook design. For the disturbance terms, �it ¼ k0PN

j¼1 wij�jt þ vit and

vit ¼ q0vit�1 þ eit, we assume li � IIDNð0; r2l0Þ and eit � IIDNð0; r2e0Þ, where the

initial value vi0 is generated from Nð0; r2e0=ð1� q20ÞÞ. Let g0 ¼ r2l0=ðr2l0 þ r2e0Þ,which takes values from the range 0 to 0.5. We use the sum r2l0 þ r2e0 to control the

signal-to-noise ratio in the design. We consider (i) a moderate signal-to-noise case

with r2l0 þ r2e0 ¼ 2 and (ii) a low signal-to-noise case with r2l0 þ r2e0 ¼ 20. We will

refer to these cases as Case 1 and Case 2, respectively. The values of other

parameters q0, s0 and k0 are varied over a range from 0 to 0.5. We consider

ðN; TÞ ¼ fð25; 7Þ; ð49; 7Þ; ð25; 12Þ; ð49; 12Þg, and report only results based on

ðN; TÞ ¼ fð25; 12Þ; ð49; 12Þg for the sake of brevity. The results for other pairs are

quite comparable to the reported ones and are available upon request.

Each Monte Carlo experiment consists of 1000 repetitions. We use the OLS

estimates to compute the eleven test statistics, namely RSr2lqks, RSr2lq, RSsk, RSr2l,

RSr2l , RSq, RSq, RS

s , RSs, RS

k and RSk. As discussed earlier, in practice, we do not

need to compute all these statistics; we do it here for comparative evaluation. The

tables and graphs are based on the nominal size of 0.05. In order to elaborate our

results systematically, we divided the results in two sections. In Sect. 6.1, we

present the Monte Carlo results for RSr2lqks, RSr2lq, RSr2l, RSr2l , RS

q and RSq, i.e., the

different test statistics for the autocorrelation and individual random effects, both in

the presence and absence of spatial parameters k and s, and in Sect. 6.2, the relating

results to testing for the spatial parameters are reported.

6.1 Monte Carlo results for tests relating to r2l and q

We investigate the performance of these tests under three setups: (i) k0 ¼ 0 and

s0 ¼ 0, (ii) k0 ¼ 0:3 and s0 ¼ 0:05, and (iii) k0 ¼ 0:05 and s0 ¼ 0:3. For (i) k0 ¼ 0

and s0 ¼ 0, we present the simulation results in Tables 3, 4, 5 and 6. In

Figs. 1, 2, 3 and 4, we consider the performance of tests under the remaining

setups: (ii) k0 ¼ 0:3 and s0 ¼ 0:05, and (iii) k0 ¼ 0:05 and s0 ¼ 0:3. Tables have

exact figures for size and power, while figures are easy to make comparison. We

first evaluate the simulation results on RSr2land RSr2l . In Tables 3, 4, 5 and 6, RSr2l

performs better than RSr2l in terms of estimated size, especially when q 6¼ 0.

However, there is a loss of power for RSr2lvis-a-vis RSr2l , and this loss gets

minimized as g0 deviates further from zero. While RSr2ldoes not sustain much loss

123


in power when q0 ¼ 0, we notice that RSr2l rejects Hb0 : r2l0 ¼ 0 too often when

r2l0 ¼ 0 and q0 6¼ 0. This unwanted rejection probabilities are due to the presence of

q0 as indicated in Corollary 1. RSr2lalso has some rejection probabilities but the

problem is less severe. Both tests have more power under Case 1 than Case 2, i.e.,

tests perform relatively better when the signal-to-noise ratio is moderate (around

0.55), instead of low.

Table 3 Empirical rejection

rates under Case 1 when s0 ¼k0 ¼ 0 and ðN;TÞ ¼ ð25; 12Þ

g0 q0 RSr2lqks RSr2lq RSr2l RSr2lRSq RSq

0.00 0.00 0.050 0.046 0.046 0.043 0.051 0.052

0.00 0.05 0.084 0.100 0.058 0.060 0.105 0.113

0.00 0.10 0.227 0.284 0.108 0.093 0.332 0.313

0.00 0.20 0.769 0.847 0.345 0.159 0.899 0.823

0.00 0.30 0.981 0.995 0.615 0.198 0.999 0.990

0.00 0.40 1.000 1.000 0.827 0.388 1.000 1.000

0.00 0.50 1.000 1.000 0.962 0.612 1.000 1.000

0.05 0.00 0.293 0.355 0.411 0.372 0.122 0.042

0.05 0.05 0.412 0.472 0.529 0.412 0.305 0.103

0.05 0.10 0.626 0.696 0.615 0.423 0.634 0.282

0.05 0.20 0.900 0.948 0.769 0.465 0.960 0.785

0.05 0.30 0.998 0.999 0.896 0.557 1.000 0.985

0.05 0.40 1.000 1.000 0.957 0.661 1.000 1.000

0.05 0.50 1.000 1.000 0.987 0.785 1.000 1.000

0.10 0.00 0.711 0.768 0.824 0.770 0.332 0.031

0.10 0.05 0.799 0.829 0.860 0.788 0.599 0.079

0.10 0.10 0.893 0.927 0.910 0.808 0.828 0.228

0.10 0.20 0.981 0.986 0.943 0.762 0.989 0.754

0.10 0.30 0.999 1.000 0.954 0.765 1.000 0.984

0.10 0.40 1.000 1.000 0.988 0.815 1.000 0.999

0.10 0.50 1.000 1.000 0.998 0.911 1.000 1.000

0.30 0.00 1.000 1.000 1.000 1.000 0.968 0.008

0.30 0.05 1.000 1.000 1.000 1.000 0.995 0.044

0.30 0.10 1.000 1.000 1.000 1.000 1.000 0.120

0.30 0.20 1.000 1.000 0.998 0.998 1.000 0.545

0.30 0.30 1.000 1.000 1.000 0.998 1.000 0.907

0.30 0.40 1.000 1.000 1.000 0.996 1.000 0.991

0.30 0.50 1.000 1.000 1.000 0.995 1.000 1.000

0.50 0.00 1.000 1.000 1.000 1.000 0.999 0.001

0.50 0.05 1.000 1.000 1.000 1.000 1.000 0.008

0.50 0.10 1.000 1.000 1.000 1.000 1.000 0.030

0.50 0.20 1.000 1.000 1.000 1.000 1.000 0.240

0.50 0.30 1.000 1.000 1.000 1.000 1.000 0.678

0.50 0.40 1.000 1.000 1.000 1.000 1.000 0.931

0.50 0.50 1.000 1.000 1.000 1.000 1.000 0.996

123


Next we consider the simulation results presented in Figs. 1 and 2 for RSr2land

RSr2l . Here, we allow k0 and s0 to deviate from zero. In terms of estimated size, it is

clear that RSr2lperforms better than RSr2l . Comparing the power properties, it is

clearly evident that the power loss gets minimized for RSr2las g0 gets larger. The

effect of the signal-to-noise ratio on the estimated power of tests can be seen by




0.00 0.00 0.050 0.046 0.038 0.041 0.062 0.051

0.00 0.05 0.137 0.171 0.078 0.065 0.200 0.177

0.00 0.10 0.425 0.536 0.207 0.095 0.615 0.522

0.00 0.20 0.975 0.989 0.537 0.152 0.996 0.980

0.00 0.30 1.000 1.000 0.847 0.294 1.000 0.999

0.00 0.40 1.000 1.000 0.977 0.534 1.000 1.000

0.00 0.50 1.000 1.000 0.996 0.809 1.000 1.000

0.05 0.00 0.541 0.630 0.688 0.625 0.219 0.040

0.05 0.05 0.695 0.758 0.773 0.619 0.609 0.150

0.05 0.10 0.863 0.912 0.848 0.597 0.889 0.524

0.05 0.20 0.998 1.000 0.941 0.666 1.000 0.984

0.05 0.30 1.000 1.000 0.990 0.726 1.000 1.000

0.05 0.40 1.000 1.000 0.998 0.844 1.000 1.000

0.05 0.50 1.000 1.000 1.000 0.928 1.000 1.000

0.10 0.00 0.940 0.958 0.975 0.967 0.577 0.038

0.10 0.05 0.967 0.985 0.991 0.962 0.883 0.130

0.10 0.10 0.989 0.994 0.990 0.959 0.981 0.432

0.10 0.20 1.000 1.000 0.998 0.959 1.000 0.971

0.10 0.30 1.000 1.000 1.000 0.957 1.000 1.000

0.10 0.40 1.000 1.000 1.000 0.972 1.000 1.000

0.10 0.50 1.000 1.000 1.000 0.991 1.000 1.000

0.30 0.00 1.000 1.000 1.000 1.000 1.000 0.010

0.30 0.05 1.000 1.000 1.000 1.000 1.000 0.052

0.30 0.10 1.000 1.000 1.000 1.000 1.000 0.248

0.30 0.20 1.000 1.000 1.000 1.000 1.000 0.909

0.30 0.30 1.000 1.000 1.000 1.000 1.000 0.998

0.30 0.40 1.000 1.000 1.000 1.000 1.000 1.000

0.30 0.50 1.000 1.000 1.000 1.000 1.000 1.000

0.50 0.00 1.000 1.000 1.000 1.000 1.000 0.000

0.50 0.05 1.000 1.000 1.000 1.000 1.000 0.013

0.50 0.10 1.000 1.000 1.000 1.000 1.000 0.074

0.50 0.20 1.000 1.000 1.000 1.000 1.000 0.603

0.50 0.30 1.000 1.000 1.000 1.000 1.000 0.975

0.50 0.40 1.000 1.000 1.000 1.000 1.000 0.999

0.50 0.50 1.000 1.000 1.000 1.000 1.000 1.000

123


comparing Figs. 1 and 2: both tests have relatively more estimated power when the

signal-to-noise ratio is moderate. Figures 1 and 2 confirm that the local presence of

the spatial dimensions do not affect RSr2ldrastically, which confirms our analytical

results. The feature of RSr2lis more or less similar when k0 ¼ 0 and s0 ¼ 0 vis-a-vis

their local departures from zero. It means that the inference on r2l0 does not depend




0.00 0.00 0.050 0.046 0.046 0.043 0.051 0.052

0.00 0.05 0.084 0.103 0.056 0.058 0.107 0.113

0.00 0.10 0.226 0.280 0.108 0.094 0.330 0.311

0.00 0.20 0.769 0.839 0.346 0.157 0.896 0.819

0.00 0.30 0.981 0.994 0.612 0.207 0.998 0.987

0.00 0.40 1.000 1.000 0.839 0.388 1.000 0.999

0.00 0.50 1.000 1.000 0.963 0.600 1.000 1.000

0.05 0.00 0.055 0.053 0.043 0.043 0.048 0.052

0.05 0.05 0.096 0.108 0.079 0.059 0.121 0.122

0.05 0.10 0.254 0.321 0.154 0.110 0.370 0.309

0.05 0.20 0.758 0.840 0.352 0.148 0.892 0.812

0.05 0.30 0.987 0.997 0.652 0.264 0.998 0.990

0.05 0.40 1.000 1.000 0.869 0.404 1.000 1.000

0.05 0.50 1.000 1.000 0.963 0.624 1.000 1.000

0.10 0.00 0.064 0.052 0.057 0.062 0.049 0.053

0.10 0.05 0.111 0.115 0.090 0.064 0.126 0.110

0.10 0.10 0.246 0.301 0.161 0.096 0.360 0.288

0.10 0.20 0.799 0.863 0.371 0.164 0.912 0.809

0.10 0.30 0.994 0.998 0.651 0.298 0.999 0.988

0.10 0.40 1.000 1.000 0.842 0.425 1.000 0.999

0.10 0.50 1.000 1.000 0.968 0.667 1.000 1.000

0.30 0.00 0.155 0.179 0.220 0.201 0.080 0.048

0.30 0.05 0.240 0.307 0.288 0.206 0.250 0.109

0.30 0.10 0.422 0.510 0.395 0.231 0.514 0.310

0.30 0.20 0.854 0.920 0.626 0.314 0.942 0.789

0.30 0.30 0.995 0.998 0.779 0.422 1.000 0.980

0.30 0.40 1.000 1.000 0.920 0.546 1.000 1.000

0.30 0.50 1.000 1.000 0.983 0.736 1.000 1.000

0.50 0.00 0.660 0.708 0.782 0.743 0.274 0.033

0.50 0.05 0.727 0.795 0.823 0.736 0.516 0.082

0.50 0.10 0.819 0.868 0.852 0.717 0.779 0.232

0.50 0.20 0.973 0.986 0.925 0.727 0.988 0.759

0.50 0.30 1.000 1.000 0.967 0.741 1.000 0.988

0.50 0.40 1.000 1.000 0.976 0.802 1.000 1.000

0.50 0.50 1.000 1.000 0.996 0.852 1.000 1.000

123


on the local presence of spatial parameters k0 and s0, confirming our result in

Corollary 5.

In a similar way, we can explain the size and power properties of RSq using

Tables 3, 4, 5 and 6, and Figs. 3, 4. From the results presented in tables, we note

that RSq has better power than RSq when rl02 ¼ 0. However, unlike RSr2l

, the power

of RSq is much closer to RSq in all cases. The real benefit of RSq is noticed when




0.00 0.00 0.050 0.046 0.038 0.041 0.062 0.051

0.00 0.05 0.137 0.174 0.078 0.065 0.200 0.178

0.00 0.10 0.425 0.536 0.207 0.098 0.616 0.518

0.00 0.20 0.973 0.990 0.533 0.158 0.996 0.981

0.00 0.30 1.000 1.000 0.848 0.287 1.000 1.000

0.00 0.40 1.000 1.000 0.977 0.545 1.000 1.000

0.00 0.50 1.000 1.000 0.996 0.816 1.000 1.000

0.05 0.00 0.055 0.042 0.041 0.049 0.043 0.046

0.05 0.05 0.161 0.196 0.106 0.067 0.212 0.171

0.05 0.10 0.481 0.568 0.218 0.101 0.659 0.553

0.05 0.20 0.977 0.990 0.585 0.174 0.997 0.976

0.05 0.30 1.000 1.000 0.892 0.307 1.000 1.000

0.05 0.40 1.000 1.000 0.980 0.568 1.000 1.000

0.05 0.50 1.000 1.000 0.999 0.812 1.000 1.000

0.10 0.00 0.069 0.060 0.067 0.064 0.057 0.047

0.10 0.05 0.161 0.201 0.141 0.075 0.233 0.151

0.10 0.10 0.491 0.590 0.280 0.112 0.688 0.555

0.10 0.20 0.987 0.995 0.606 0.199 1.000 0.986

0.10 0.30 1.000 1.000 0.895 0.343 1.000 1.000

0.10 0.40 1.000 1.000 0.991 0.585 1.000 1.000

0.10 0.50 1.000 1.000 0.999 0.830 1.000 1.000

0.30 0.00 0.245 0.308 0.355 0.332 0.112 0.050

0.30 0.05 0.407 0.524 0.501 0.333 0.411 0.151

0.30 0.10 0.737 0.819 0.642 0.361 0.828 0.518

0.30 0.20 0.995 0.998 0.872 0.449 1.000 0.986

0.30 0.30 1.000 1.000 0.953 0.550 1.000 0.999

0.30 0.40 1.000 1.000 0.998 0.734 1.000 1.000

0.30 0.50 1.000 1.000 1.000 0.906 1.000 1.000

0.50 0.00 0.917 0.943 0.964 0.954 0.510 0.025

0.50 0.05 0.943 0.967 0.972 0.920 0.835 0.141

0.50 0.10 0.987 0.993 0.987 0.919 0.980 0.476

0.50 0.20 1.000 1.000 0.998 0.924 1.000 0.972

0.50 0.30 1.000 1.000 0.999 0.925 1.000 1.000

0.50 0.40 1.000 1.000 1.000 0.952 1.000 1.000

0.50 0.50 1.000 1.000 1.000 0.980 1.000 1.000

123


q0 ¼ 0 but g0 [ 0; the performance of RSq is remarkable when the signal-to-noise

ratio is low in Tables 5 and 6. Even when there is local presence of the parameters

k0 and s0, RSq is size-robust as shown in Fig. 4. In other words, RSq performs well

and does not reject Hc0 : q0 ¼ 0 when q0 is indeed zero even for large values of g0.

The non-centrality parameter of RSq, as shown in Corollary 2 is independent of any

nuisance parameters, which explains the robust performance of RSq. On the other

hand RSq rejects the null too often when g0 is large. This can be easily observed

from k2 in Corollary 2, which is a function of r2l0.Results from the joint statistics RSr2lqks and RSr2lq are informative when we

accept the respective null hypotheses Ha0 : r2l0 ¼ q0 ¼ k0 ¼ s0 ¼ 0 and

H0 : r2l0 ¼ q0 ¼ 0. However, if the corresponding null hypothesis is rejected we

need to decompose RSr2lqks and RSr2lq to extract exact source(s) of misspecification.

However, overall they have good power as shown in Tables 3, 4, 5 and 6. These

results are consistent with Bera et al. (2001) and also Montes-Rojas (2010).

However, we differ from each of them in our basic model framework, which is more

general than both Bera et al. (2001) and Montes-Rojas (2010).

In Table 7, we compare our results on RSq with the conditional LM test

suggested by Baltagi et al. (2007, pp. 8–9) for the null hypothesis H0 : q0 ¼ 0 in the

presence of r2l0 and k0. Here, RSqjr2lk refers to the one dimensional conditional LM

test as derived in Baltagi et al. Baltagi et al. (2007). The rejection probabilities for

RSqjr2lk are those reported in Baltagi et al. (2007, Table 3). As noted before, RSq is

computed by using the simple OLS estimates, whereas the computation of RSqjr2lk

requires maximum likelihood estimation of k0 and r2l0. Results reported in Table 7

further supports our findings, i.e., on the one hand the performance of our adjusted

RS statistic is very similar to the one directional conditional LM test; on the other,

our adjusted RS test is simple to compute than the conditional LM test.

6.2 Monte Carlo results for tests relating to s and k

In this section, we consider the simulation results on the parameters of spatial

dimensions. To explore the performance of these tests, we performed the Monte

Carlo study under a moderate signal-to-noise ratio for the following cases: (i) g0 ¼q0 ¼ 0 (this case is exactly similar to Anselin et al. (1996), and our results are

comparable to their findings), (ii) g0 ¼ 0:05 and q0 ¼ 0:3, and (iii) g0 ¼ 0:3 and

q0 ¼ 0:05. The results of the last two cases are comparable to the Monte Carlo

results of Baltagi et al. (2007) and Baltagi and Liu (2008).

In Tables 8 and 9, we report the estimated rejection rates of RSk, RSk, RSs , RSs,

RSr2lqks, and RSks when g0 ¼ q0 ¼ 0. In the local presence of s0, RSk reports

rejection frequencies that are very close to the nominal value of 0.05 whereas RSk issignificantly over-sized. In both tables, the size robustness of RSk is clearly evident

as its rejection rates are close to 0.05 even when s0 ¼ 0:5, i.e., under strong

presence of s0. In other words, RSk does its job very well, even better than what it is

designed to do for. However, the rejection rates of RSk are large in the presence of

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s0 when k0 is actually equal to zero. This confirms our result that RSk is robust to

local and as well as global misspecification, while the test results of RSk can be very

misleading in the presence of the nuisance parameters (see the k2 measure in

Corollary 3). In terms of power, RSk is trailing just behind RSk as can be clearly

seen from Tables 8 and 9.

From Tables 8 and 9, it also is evident that RSs is size robust for the local

presence of k0. For k0 [ 0 and s0 ¼ 0, the size distortions in RSs are very large,

whereas they are much smaller in the case of RSs . This unwanted rejection

probabilities of RSs is due to the non-centrality parameter, which depends on k0 asshown by the k2 measure in Corollary 4. As we mentioned earlier, RSs is designedto be robust only under local misspecification, i.e., for low values of k0. From that

point of view, it does a good job; its performance only deteriorates as k0 gets much

Fig. 1 Estimated size and power under Case 1 when ðN; TÞ ¼ ð25; 12Þ

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larger. From both tables, we also note that an increase in k0 enhances the rejection

probabilities of RSs when s[ 0. This is due to the presence of k0 in the non-

centrality parameter of RSs (see Corollary 4).

The simulation results under the remaining cases, (ii) g ¼ 0:05 and q ¼ 0:3, and(iii) g ¼ 0:3 and q ¼ 0:05, are reported in Figs. 5 and 6. The size properties of RSkare much better than its unadjusted counterpart in the local presence of other three

parameters g0, q0 and s0. The power of RSk is slightly less than that of RSk in both

cases. Figures 5 and 6 clearly show that there is almost no size distortions in RSk ass0 ranges from 0 to 0.5. On the other hand, the size distortions in RSk increase as s0gets larger. Again, these patterns can be explained by using the non centrality

parameter in the presence of nuisance parameters as in Corollary 3. These results

reiterate our earlier results presented in Tables 8 and 9 under the absence of the


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random and error autocorrelation effects, i.e, when g0 ¼ q0 ¼ 0. These experimen-

tal results provide further support for our analytical findings.

Finally, we discuss the simulation results in Figs. 5 and 6 for RSs and RSs under(ii) g0 ¼ 0:05 and q0 ¼ 0:3, and (iii) g0 ¼ 0:3 and q0 ¼ 0:05. The estimated sizes of

RSs are very close to the nominal value of 0.05 under the local presence of k0. Incontrast, the estimated size of RSs approaches to 1 when k0 approaches to 0.5. Theserejection probabilities can be explained by the non-centrality parameters stated in

Corollary 4. The power of RSs trails behind RSs, but becomes close to each other for

larger values of s0. These results are in-line with our findings from Tables 8 and 9

when g0 and q0 were zero.

One important thing to note is that these one-dimensional robust tests are more

meaningful not only than their marginal counterparts but also than the joint tests,


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rates under Case 2 when k0 ¼ 0

and ðN;TÞ ¼ ð25; 12Þq0 g0 ¼ 0 g0 ¼ 0:2 g0 ¼ 0:5

RSq RSqjr2lk RSq RSqjr2lk RSq RSqjr2lk

0.0 0.052 0.062 0.042 0.051 0.033 0.066

0.2 0.819 0.815 0.825 0.816 0.759 0.848

0.4 0.999 1.000 0.999 0.990 1.000 0.982

0.6 1.000 1.000 1.000 1.000 1.000 1.000

0.8 1.000 1.000 1.000 1.000 1.000 1.000

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RSr2lqks, RSr2lq, and RSks. The joint tests are only optimal when r2l0 ¼ q0 ¼ k0 ¼s0 ¼ 0 holds, and they fail to identify the exact source of misspecification. This is

evident from our results presented in Tables 3, 4, 5, 6, 7, 8 and 9. In addition, as

stated before, our robust tests not only provide intuitive results which can be

explained analytically, but are also easy to compute (they are all based on the OLS


rates under Case 2 when g0 ¼q0 ¼ 0 and ðN; TÞ ¼ ð25; 12Þ

s0 k0 RSr2lqks RSks RSk RSk RSs RSs

0.00 0.00 0.050 0.056 0.039 0.055 0.048 0.059

0.00 0.05 0.056 0.067 0.078 0.052 0.074 0.055

0.00 0.10 0.144 0.182 0.218 0.124 0.135 0.055

0.00 0.20 0.487 0.597 0.681 0.367 0.438 0.067

0.00 0.30 0.898 0.946 0.972 0.701 0.824 0.066

0.00 0.40 0.995 1.000 1.000 0.926 0.973 0.074

0.00 0.50 1.000 1.000 1.000 0.987 0.999 0.100

0.05 0.00 0.072 0.087 0.069 0.031 0.130 0.086

0.05 0.05 0.167 0.222 0.216 0.074 0.240 0.121

0.05 0.10 0.318 0.425 0.470 0.134 0.442 0.096

0.05 0.20 0.729 0.817 0.881 0.391 0.768 0.096

0.05 0.30 0.977 0.991 0.993 0.705 0.968 0.139

0.05 0.40 0.999 1.000 1.000 0.918 0.998 0.131

0.05 0.50 1.000 1.000 1.000 0.981 1.000 0.151

0.10 0.00 0.254 0.321 0.226 0.041 0.426 0.242

0.10 0.05 0.400 0.498 0.416 0.074 0.585 0.285

0.10 0.10 0.580 0.697 0.685 0.122 0.763 0.238

0.10 0.20 0.908 0.954 0.964 0.389 0.947 0.249

0.10 0.30 0.998 1.000 1.000 0.707 0.995 0.262

0.10 0.40 1.000 1.000 1.000 0.925 1.000 0.241

0.10 0.50 1.000 1.000 1.000 0.984 1.000 0.247

0.30 0.00 0.997 0.999 0.974 0.052 0.999 0.986

0.30 0.05 0.999 1.000 0.996 0.070 1.000 0.974

0.30 0.10 1.000 1.000 0.999 0.097 1.000 0.980

0.30 0.20 1.000 1.000 1.000 0.347 1.000 0.949

0.30 0.30 1.000 1.000 1.000 0.612 1.000 0.937

0.30 0.40 1.000 1.000 1.000 0.878 1.000 0.905

0.30 0.50 1.000 1.000 1.000 0.965 1.000 0.828

0.50 0.00 1.000 1.000 1.000 0.049 1.000 1.000

0.50 0.05 1.000 1.000 1.000 0.037 1.000 1.000

0.50 0.10 1.000 1.000 1.000 0.055 1.000 1.000

0.50 0.20 1.000 1.000 1.000 0.201 1.000 1.000

0.50 0.30 1.000 1.000 1.000 0.440 1.000 1.000

0.50 0.40 1.000 1.000 1.000 0.688 1.000 1.000

0.50 0.50 1.000 1.000 1.000 0.877 1.000 0.991

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estimates) relative to the one-dimensional conditional LM and LR tests. Moreover,

one can easily derive our robust test statistics by using the non-robust ones as shown

in Corollary 7.


rates under Case 2 when g0 ¼q0 ¼ 0 and ðN; TÞ ¼ ð49; 12Þ

s0 k0 RSr2lqks RSks RSk RSk RSs RSs

0.00 0.00 0.050 0.044 0.042 0.052 0.051 0.057

0.00 0.05 0.076 0.097 0.128 0.090 0.082 0.047

0.00 0.10 0.238 0.319 0.392 0.216 0.242 0.067

0.00 0.20 0.787 0.884 0.941 0.653 0.732 0.050

0.00 0.30 0.993 0.998 0.999 0.953 0.978 0.062

0.00 0.40 1.000 1.000 1.000 0.997 1.000 0.068

0.00 0.50 1.000 1.000 1.000 1.000 1.000 0.119

0.05 0.00 0.147 0.177 0.136 0.061 0.230 0.147

0.05 0.05 0.280 0.368 0.388 0.086 0.435 0.130

0.05 0.10 0.592 0.688 0.747 0.220 0.706 0.146

0.05 0.20 0.968 0.989 0.995 0.655 0.959 0.157

0.05 0.30 1.000 1.000 1.000 0.935 0.999 0.140

0.05 0.40 1.000 1.000 1.000 0.998 1.000 0.146

0.05 0.50 1.000 1.000 1.000 1.000 1.000 0.183

0.10 0.00 0.476 0.593 0.400 0.042 0.696 0.435

0.10 0.05 0.721 0.819 0.729 0.083 0.878 0.428

0.10 0.10 0.908 0.954 0.952 0.209 0.962 0.419

0.10 0.20 0.998 0.999 1.000 0.657 0.998 0.414

0.10 0.30 1.000 1.000 1.000 0.953 1.000 0.389

0.10 0.40 1.000 1.000 1.000 0.998 1.000 0.378

0.10 0.50 1.000 1.000 1.000 1.000 1.000 0.380

0.30 0.00 1.000 1.000 1.000 0.047 1.000 1.000

0.30 0.05 1.000 1.000 1.000 0.077 1.000 1.000

0.30 0.10 1.000 1.000 1.000 0.172 1.000 0.997

0.30 0.20 1.000 1.000 1.000 0.591 1.000 0.998

0.30 0.30 1.000 1.000 1.000 0.920 1.000 0.999

0.30 0.40 1.000 1.000 1.000 0.997 1.000 0.992

0.30 0.50 1.000 1.000 1.000 1.000 1.000 0.977

0.50 0.00 1.000 1.000 1.000 0.035 1.000 1.000

0.50 0.05 1.000 1.000 1.000 0.040 1.000 1.000

0.50 0.10 1.000 1.000 1.000 0.094 1.000 1.000

0.50 0.20 1.000 1.000 1.000 0.410 1.000 1.000

0.50 0.30 1.000 1.000 1.000 0.792 1.000 1.000

0.50 0.40 1.000 1.000 1.000 0.954 1.000 1.000

0.50 0.50 1.000 1.000 1.000 0.995 1.000 1.000

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7 Conclusion

In this paper, we proposed the ‘‘robust’’ Rao’s score (RS) test for random effect,

serial correlation, spatial error and spatial lag dependence in the context of a spatial

panel data model. These tests are robust in the sense that they are asymptotically

valid in the (local) presence of nuisance parameters. The computation of our

suggested tests requires only the OLS estimates. After one has the standard RS tests

for each parameter, our robust tests require very little extra computation. Thus,

practitioners can identify specific direction(s) to reformulate the basic model

without going through any complex estimation. Our empirical illustration in the


123


context of the convergence theory of income demonstrates the usefulness of our

proposed tests, in particular, to identify the most reasonable departures from the

basic panel regression model. We also investigated the finite sample size and power

properties of our proposed tests through an extensive Monte Carlo study, and

compared them with the performance of some of the available tests. Our tests

perform very well in finite sample and compare favorably to other tests that require

explicit estimation of nuisance parameters. Also, though our methodology is

developed only for local misspecification, our results from simulation experiments

show that our tests perform quite well for non-local departures in certain cases.


123


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https://doi.org/10.2307/1906935

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps

and institutional affiliations.

Affiliations

Anil K. Bera1 • Osman Dogan1 • Suleyman Taspınar2 • Monalisa Sen1

& Osman Dogan

[email protected]

Anil K. Bera

[email protected]

Suleyman Taspı[email protected]

Monalisa Sen

[email protected]

1 Department of Economics, University of Illinois at Urbana-Champaign, Champaign, USA

2 Department of Economics, Queens College, The City University of New York, New York, USA

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http://orcid.org/0000-0001-7324-9454

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