AN EXTENSION OF THE J‐TEST TO A SPATIAL PANEL DATA FRAMEWORK

An Extension of the J -test to a spatial panel dataframework

Harry H. KelejianUniversity of Maryland

College Park, MD 20742, USA

Gianfranco Piras∗

West Virginia UniversityMorgantown, WV 26506, USA

Abstract

Kelejian (2008) extended the J-test procedure to a spatial framework. Al-though his suggested test was computationally simple and intuitive, it did notuse the available information in an efficient manner. Kelejian and Piras (2011)generalized and modified Kelejian’s test to account for all the available informa-tion. However, neither Kelejian (2008) nor Kelejian and Piras (2011) considereda panel data framework.

In this paper we generalize these earlier works to a panel data framework withfixed effects and additional endogenous variables. We give theoretical, as well asMonte Carlo results relating to our suggested tests. An empirical application ona crime model for North Carolina is also estimated.

JEL classification: C01, C12Key Words: Spatial Models, General Specifications, Non-nested J-test

1 Introduction

The J-test is a procedure for testing a null model against non-nested alternatives.1 Asdescribed in Kelejian and Piras (2011), the J-test is based on whether or not predictionsof the dependent variable based on the alternative models add significantly to theexplanatory power of the null model.

∗Piras is the contacting author: Regional Research Institute, West Virginia University, 886 ChestnutRidge Road, Room 510, P.O. Box 6825, Morgantown, WV 26506-6825; E-mail: [email protected]

1There is, of course, a large literature relating to the J-test. For example, see Davidson andMacKinnon (1981); MacKinnon et al. (1983); Godfrey (1983); Pesaran and Deaton (1978); Dastoor(1983); Pesaran (1974, 1982); Delgado and Stengos (1994), and the reviews given in Greene (2003,pp.153-155, 178-180) and Kmenta (1986, pp 593-600). A nice overview of issues relating to non-nestedmodels is given in Pesaran and Weeks (2001).

1

Kelejian (2008) extended the J-test procedure to a spatial framework but the sug-gested test was not based on all of the available information. This was pointed out byKelejian and Piras (2011) who, among other things, generalized Kelejian’s assumptions.However, neither Kelejian (2008) nor Kelejian and Piras (2011) considered a panel dataframework. This is unfortunate because a great many studies in recent years have beenin a panel data framework.2

In this paper we generalize these earlier works on the J-test to a fixed effects paneldata framework. We specify a null model which contains spatial lags in both thedependent variable and the disturbance term, as well as additional (other than thespatial lag) endogenous variables. We allow for G alternative models which can also,but need not, have such “complications”. The error terms in these alternative modelsare specified in a general way so that spatial correlation of various sorts, as well asgeneral patterns of heteroskedasticity are special cases. However, if there is a singlealternative model which is identical to the null except that it has a random effectstructure, the J-test framework for discriminating between these two models will haveweak power and so should not be used.3

As in Kelejian and Piras (2011) we show that, given a critical assumption, the fullinformation J-test in a panel is computationally simple, and indeed, simpler than theones suggested in Kelejian (2008). We also demonstrate that this “crucial” assumptionwould typically be satisfied in most spatial models. Finally, we design an extensiveMonte Carlo experiment and give results that suggest that our proposed J-tests havedecent power, and proper size even for relatively small samples.

Finally, an empirical application to a crime model for North Carolina is also con-sidered. The results of the J-test applied to this situation suggest that counties onlytake into account policies implemented by their immediate neighbors.

In Section 2 we specify the null and alternative models. Section 3 contains a discus-sion of the J-test. In Section 4 we introduce the Monte Carlo model and present theresults in Section 5. The empirical application is discussed in Section 6. Conclusionsare given in Section 7. Technical details are relegated to the appendix.

2 The null and alternative models

The null model

Let eT be a T × 1 vector of unit elements. Consider the null panel data model with

2For example, see Anselin et al. (2008); Kapoor et al. (2007); Baltagi et al. (2007b, 2003); Baltagiand Liu (2008); Baltagi et al. (2007a, 2009); Piras (2013); Debarsy and Ertur (2010); Elhorst (2003);Elhorst and Freret (2009); Elhorst (2008, 2009, 2010); Elhorst et al. (2010); Lee and Yu (2010c,a,b,d);Mutl and Pfaffermayr (2011); Pesaran and Tosetti (2011); Yu and Lee (2010); Yu et al. (2008); Parentand LeSage (2010)

3Mutl and Pfaffermayr (2011) suggest and give large sample results for a Hausman test to discrim-inate between these two models.

2

fixed effects

y = Xβ0 + λ0(IT ⊗W )y + Y d0 + (eT ⊗ IN)µ0 + u (1)

≡ Zγ0+(eT ⊗ IN)µ0 + u; Z=(X, (IT ⊗W )y, Y )

γ0 = (β′0, λ0, d′0)′; u = ρ0(IT ⊗W )u+ ε

where y is the NT × 1 vector of values of the dependent variable corresponding to Ncross sectional units over T time periods, X is an NT × k matrix of observations onexogenous variables, W is an N ×N nonstochastic weighting matrix; Y is an NT × hmatrix of observations on h endogenous variables, µ0 is the N ×1 fixed effects vector; uis the corresponding disturbance vector which is specified as a panel SAR process in thethird line of (1) where ε is the random innovation vector; β0 and d0 are, respectively, k×1and h×1 parameter vectors, and λ0 and ρ0 are scalar parameters. To avoid unnecessarycomplications, we have assumed that the weighting matrix in the regression model isthe same as the one in the error process. This assumption is typically made in practice.Our results can easily be extended to the case in which these two weighting matricesare not equal.

We allow for triangular arrays but do not index the variables in (1) with the samplesize in order to simplify the notation. We also assume that the system determiningthe endogenous variables in Y is not known. Therefore, we do not consider maximumlikelihood estimation. In a typical limited information setting underlying instrumentalvariable estimation, we assume that the researcher has observations on r ≥ h exogenousvariables that appear in that unknown system. For future reference, let the NT × rmatrix of observations on these r exogenous variables be S.

Using evident notation, let the elements of ε be εit, i = 1, ..., n; t = 1, ..., T. Then,we assume the εit is i.i.d. over both i and t with mean and variance (0, σ2

ε). Thisintuitive specification is more than adequate to understand our presentation below, butit does not allow for triangular arrays. A formal specification which does is given asAssumption 1 in Kapoor et al. (2007) In order for the model to be complete, we alsoassume that (IN − aW ) is nonsingular for all |a| < 1.

Let Q0 = ((IT − JTT

) ⊗ IN), JT = eT e′T , and note that Q0(eT ⊗ IN) = 0. Thus,

pre-multiplying the second and third lines in (1) by Q0 yields

Q0y = Q0Zγ0 +Q0u (2)

Q0u = ρ0(IT ⊗W )Q0u + Q0ε

since Q0(IT ⊗W ) = (IT ⊗W )Q0. Finally, consistent with the spatial Cochrance-Orcutt

procedure, let

y∗(ρ0) = [INT−ρ0(IT ⊗W )]Q0y (3)

Z∗(ρ0) = [INT−ρ0(IT ⊗W )]Q0Z

and note from (2) thaty∗(ρ0) = Z∗(ρ0)γ0 +Q0ε (4)

3

The G alternative models under H1

We assume the researcher believes that the true alternative to the null is one ofJ = 1, ..., G models, namely

H1,J : yt = Pt,JβJ + λJMJ yt + Yt,JdJ + µJ + ψt,J ; J = 1, ..., G; t = 1, ..., T (5)

where yt is the N ×1 vector of observations corresponding to the dependent variable attime t, PJ,t is the N × kJ matrix of observations on the exogenous variables in the J th

model at time t, MJ is the corresponding weighting matrix, Yt,J is an N ×hJ matrix ofobservations on hJ endogenous variables at time t, µJ is a fixed effects vector, ψt,J isthe corresponding disturbance vector, etc. As for the null model, we assume that theonly information on the system determining the elements of Yt,J are observations on rJexogenous variables. At this point, note that unlike for the null model, we have specifythese alternative models for each time period t.

Stacking the model in (5) over t = 1, ..., T , and using evident notation we have

H1,J : y = PJβJ + λJ(IT ⊗MJ)y + YJdJ + (eT ⊗ IN)µJ + ψJ (6)

≡ ZJ γJ + (eT ⊗ IN)µJ + ψJ ; ZJ = [PJ , (IT ⊗MJ)y, YJ ]; γJ = (β′J , λJ , d′J)′

Multiplying the second line in (6) across by Q0 we have

Q0y = Q0ZJ γJ +Q0ψJ ; J = 1, ..., G (7)

since Q0(IT ⊗MJ) = (IT ⊗MJ)Q0. For future reference, let SJ be the NT × rJ matrixof observations on the rJ exogenous variables which appear in the system determiningthe elements of Y under H1,J .

Our assumptions concerning the disturbance terms in the alternative models arequite general. Specifically, we only assume

E(ψJ |H1,J ,Φ1,J) = 0 and E(ψJψ′J |H1,J ,Φ1,J) = ΩψJ

(8)

where Φ1,J = (PJ ,MJ , SJ), and the row and column sums of ΩψJand Ω−1ψJ

are uniformlybound in absolute value. This assumption is consistent with general patterns of spatialcorrelation and heteroskedasticity. It is also typically assumed in large sample analysesof spatial models- see, e.g., Kelejian and Prucha (1999, 2004, 2007), Yu et al. (2008),and Mutl and Pfaffermayr (2011).

3 The form of the J-test

The augmented equation

Our J-test is based on augmenting (4) with the G predictions of y∗(ρ0) based onthe G alternative models, namely

E(y∗(ρ0)|H1,J , INFOJ) = [INT−ρ0(IT ⊗W )] [Q0E(y|H1,J , INFOJ)]; J = 1, ..., G(9)

4

where INFOJ is the information set the predictions are based on. In essence, our testof H0 versus H1 is a test of the joint significance of the G augmenting variables.

We consider two information sets and so there are two forms of our augmentedmodel. The first set is just Φ1,J . Let yi,t be the ith value of yt and let Yi,t,J be theith row of Yt,J . Then, the second information set is a full information set, say Φ2,J ,which relates to the prediction of each yi,t based on the J th alternative model. Let Mi,J

be the ith row of MJ . Then, as discussed in more detail in the appendix, our secondinformation set, say Φ2,J , augments Φ1,J for each yi,t with Mi,Jyt and Yi,t,J which wouldbe regressors in the J th model under H1 for yi,t - see (5). Since weighting matrices havezeroes on the diagonal, Mi,Jyt does not include yi,t but it is obviously correlated with

the error term, as would be Yi,t,J in the J th model. Let Q0y(1)J = Q0E(y|H1,J ,Φ1,J)

and, in (10) below, let Q0y(2)J = Q0E(y|H1,J ,Φ2,J)−Q0ηJ . Again, as demonstrated in

the appendix

Q0E(y|H1,J ,Φ1,J) = [IT ⊗ (IN − λJMJ)−1][Q0PJβJ +Q0E(YJ |H1,J ,Φ1,J)dJ ](10)

Q0E(y|H1,J ,Φ2,J) = Q0PJβJ + λJ(IT ⊗MJ)Q0y + YJdJ +Q0ηJ

where ηJ = E(ψJ |H1,J ,Φ2,J) 6= 0, but as shown in the appendix, E(ηJ |H1,J ,Φ1,J) = 0.Under reasonable conditions, we show in the appendix that if H1,J is the only alter-

native under H1, i.e. G = 1, the term involving ηJ in (10) is asymptotically negligibleeven if it is observed, and so Q0ηJ can be ignored. In addition, a demonstration virtu-ally identical to that in Kelejian and Piras (2011) will show that the power of the J-testbased on the predictor Q0PJ,tβJ+λJ(IT⊗MJ)Q0y+YJdJ is asymptotically equivalent tothe power of the J-test based on [IT ⊗(IN−λJMJ)−1][Q0PJβJ +Q0E(YJ |H1,J ,Φ1,J)dJ ],again when G = 1. This predictor corresponds to the mean of the dependent variablevia the reduced form of (6). Our Monte Carlo results are consistent with these asymp-totic results even if G > 1.0. Since the predictor on the second line of (10), minus theterm Q0ηJ , does not involve an inverse, we suggest its use.

LetY

(s)J = [INT−ρ0(IT ⊗W )]Q0y

(s)J , s = 1, 2; J = 1, ..., G (11)

Our suggested augmented equation is based on y(2)J , J = 1, ..., G :

y∗(ρ0) = Z∗(ρ0)γ0 + Y (2)α +Q0ε (12)

Y (2) = (Y(2)1 , ..., Y

(2)G ); α′ = (α1, ..., αG)

where α is aG×1 parameter vector. For future reference we define Y (1) = (Y(1)1 , ..., Y

(1)G ).

To implement the J-test, the parameters ρ0, βJ , dJ , and λJ defining Y(2)J in (11) must

be estimated, J = 1, ..., G. Our Monte Carlo results suggest that even in moderatelysized samples, the power of the J-test is not very sensitive to the efficiency with whichthese parameters are estimated as long as, under H1,J , these estimators are consistent.This implies that in implementing the J-test, the alternative models can simply beestimated by an IV procedure which need not to account for particular “complications”

5

the researcher may assume for the error term. The parameter ρ0 is estimated in termsof the null model, as described below.

The necessary calculations and the test

To implement the J-test, the null model, and all G alternative models must be esti-mated in a preliminary, but consistent fashion. Given these estimations, the augmentedmodel must then be estimated. The instrument matrices we use to estimate the nullmodel, and the alternative models are, respectively,

H(0) = [Q0X,Q0S, (IT ⊗W )Q0X, (IT ⊗W 2)Q0X]LI (13)

H(J) = [Q0PJ , Q0SJ , (IT ⊗MJ)Q0PJ , (IT ⊗M2J)Q0PJ ]LI

J = 1, ..., G.

where LI denotes the linearly in dependent columns of the matrix in brackets. Theaugmented model is estimated using the union of these instrument matrices

H(A) = [H(0), H(1), ..., H(G)]LI (14)

Step 1: We follow Mutl and Pfaffermayr (2011), Kapoor et al. (2007), and Piras(2013) in estimating the fixed effects null model in (2) by 2SLS based on H(0) and usethe residuals to obtain the GMM estimator of ρ0, say ρ0. This estimator is based onthe first three equations of the GMM procedure in Kapoor et al. (2007).

Step 2: Estimate the regression parameter vector γJ of the alternative models in(7) by any consistent method under H1,J . One such method would be 2SLS based onthe instrument matrix H(J).

Step 3: Let γJ , J = 1, ..., G be a consistent estimator of γJ . Let Y (2) = (Y(2)1 , ..., Y

(2)G )

where Y(2)J is identical to Y

(2)J in (11) except that ρ0, βJ ,dJ , and λJ are replaced by

ρ0, βJ , dJ , and λJ , J = 1, ..., G. Similarly, let y∗(ρ0) and Z∗(ρ0) be identical to y∗(ρ0)and Z∗(ρ0) in (3) except that ρ0 is replaced by ρ0. Then, the empirical form of ouraugmented model is

y∗(ρ0) = Z∗(ρ0)γ0 + Y α + ζ (15)

= Fδ + ζ

where F = (Z∗(ρ0), Y ), and δ′ = (γ′0, α′), and ζ is an error term.

Step 4: Let F = H(A)(H′(A)H(A))

−1H ′(A)F where the instrument matrix H(A) is

defined in (14). Then, our test is based on the 2SLS estimator of δ in (15), namely,

δ = (F ′F )−1F ′y∗(ρ0) (16)

Assuming the standard conditions in Mutl and Pfaffermayr (2011) with their regressormatrix expanded to include Y , and those in Kapoor et al. (2007), ρ0 is consistent and

(NT )1/2[δ − δ] D→ N(0, σ2ε p lim

N→∞(NT )[Γ(H ′(A)H(A))

−1 Γ′]) (17)

Γ = (F ′F )−1[F ′H(A)]

6

Small sample inferences can be based on the approximation

δ.∼ N(δ, σ2

εΓ(H ′(A)H(A))−1 Γ′) (18)

where4

σ2ε =

1

N(T − 1)[y∗(ρ0)− F δ]′[y∗(ρ0)− F δ] (19)

Our test of H0 against H1 is to reject H0 at the 5% level if

α′[V C−1α ]α > χ2G(.95)

where α is the last G elements of δ and V Cα is the lower diagonal G × G block ofσ2εΓ(H ′(A)H(A))

−1 Γ′.

4 The Monte Carlo model

The experimental design for the Monte Carlo simulation is based on a format extensivelyused in studies on spatial panel regression models- e.g., (Kapoor et al., 2007; Baltagiet al., 2003, 2007b; Debarsy and Ertur, 2010).

Specifically, we generate two sets of data corresponding to two regular grids ofdimension 7 × 7 and 10 × 10, leading to sample sizes of, respectively, N = 49 andN = 100. We only consider one value of the time dimension, namely T = 4. For eachsample size we construct three row normalized weights matrices. Following Kelejianand Prucha (1999), the first of these three matrices (W1) is defined in a circular worldand is generally referred to as “5 ahead and 5 behind” spatial weights matrix. Oursecond matrix (W2) is a distance matrix based on the ten nearest neighbors.5 Finally,the third matrix (W3) is a contiguity matrix based on the rook criterion (i.e. onlyborders but not vertex).

For each sample size, we design six sets of experiments. In the first two sets, the nulland alternative models only differ in terms of the weighting matrix. In the third set, thenull and alternative models differ only in terms of the regressor matrix; in the fourthset they differ in terms of both their regressors and weighting matrices. In the last twosets of experiments, we consider two models under the alternative. In the fifth set, thenull and alternative models only differ in terms of the weighting matrix employed. Inthe sixth set, they differ in terms of both the regressors and the weighting matrix.

4Note that the expression for σ2ε is divided by 1

N(T−1) . This is actually in line with the findings

of Lee and Yu (2010a). In a different context related to maximum likelihood estimation they suggesta transformation - i.e., based on the orthonormal matrix of the eigenvectors of JT to avoid lineardependence of the disturbances over the time dimension. Furthermore, when the model is specifiedonly in terms of individual effects, both the transformation and the direct approach lead to the samecoefficients estimate except for σ2

ε. In fact, the estimation of σ2ε from the direct approach will be T−1

Ttimes the estimate from the transformation approach.

5We used euclidean distance to determine the ten nearest neighbors.

7

Using evident notation, in all of the experiments the null model is of the form:

y =λ(IT ⊗W )y +Xβ + u (20)

u =ρ(IT ⊗W )u+ ε

where, for simplicity, we assume that the weighting matrix in the regression model isthe same as the one in the disturbance process.6

We consider two distributions for ε. In the first case the elements of ε are i.i.d.N(0, 1). As in Kelejian and Prucha (1999), the second distribution considered is anormalized version of the log-normal, henceforth abbreviated as NLogN . In this casethe elements of ε are specified as

εi = [exp(ξi)− exp(.5)]/[exp(2)− exp(1)].5 (21)

where ξi is i.i.d.N(0, 1). This normalization implies that ε is i.i.d.(0, 1) but the distri-butions in (21) is not symmetric.7

In all experiments, except those in the fourth set, the alternative models are definedonly in terms of the first line of (20).8

There are two regressor matrices one for the null (X0) and one for the alternativemodel (X1).

9 The regressor matrix X0 is taken as X0 = (x0, x1). Following Debarsy andErtur (2010), the (i, t)th value of x0, as well as that of x1, is an independent draw fromN(µt, 1) where µt is an independent draw from U(0, 1), t = 1, ..., 4; i = 1, ..., N. Theregressor matrix X1 is taken as X1 = (z1, z2), where the NT values of z1 are generatedin the same way as x0 (and x1) and

z2 = ax0 + ξ (22)

where ξ = N(0, INT ) and a = .5. As in Kelejian and Piras (2011), we choose this valueof a because it leads to a correlation between z1 and x0 of, approximately 0.5. In allexperiments, once generated, the values of the regressors are held fixed in the MonteCarlo trials. Finally, the elements of β are taken as 1.0 or 0.5 These values of theparameters lead to the ratio of the variance of Xβ to the sum of the variance of Xβand the variance of the error term of, approximately, 0.35 and 0.68. We refer to thisratio as R2 in our tables in the next section.

In all sets of experiments, we consider six values for λ, namely−0.6,−0.4,−0.2, 0.2, 0.4,and 0.6; and two values for ρ, namely −0.4, 0.4.

6Although restrictive, this assumption is generally made in many spatial econometrics applications(e.g. Donovan et al., 2007; Arraiz et al., 2010; Piras and Lozano-Gracia, 2012).

7Since the results for the NLogN are virtually identical, they are not reported in the paper.However, they can be obtained from the authors.

8In the fourth set of experiments we also consider the same two distributions for the disturbanceterms. However, unlike for the other experiments, the alternative model has the same structure as in(20).

9The use of these matrices in the various sets of experiments is described in the tables.

8

For all experiments, 2,000 replications are performed. This is, roughly, the numberof replications needed to obtain a 95% confidence interval of length .019 on the size ofa test statistic- i.e., an estimate of the size of the test in the tables below would beviewed as being significantly different than the .05 theoretical level if it is not in theinterval (.041, .060).

5 Monte Carlo results

Our Monte Carlo results are summarized in Tables 1-4. These tables give the frequencyof rejection of the null hypothesis at the 5% level. These figures relate to the estimatedsize of the test, as well as to its estimated power. As already mentioned, all the resultsare based on the disturbance distribution εn = N(0, In). Except for Table 3 (discussedlater), the results in the first four columns of the tables corresponds to the sample sizeof N = 49 observations, whereas the results in the remaining columns relate to largersample size (i.e. N = 100). Additionally, the top part of the tables refer to the lowervalue of the R2 statistic while the bottom part to the higher R2. Results are given foreach of the two predictors y(1) and y(2).

Due to space limitation, we only report on some experiments. Since experimentsone and two are similar (i.e. only the spatial weighting matrix are different betweenthe null and alternative model in both experiments) we only show the complete set ofresults (for the normal distribution of the error term) for the first experiment (Table1). The results reported in Tables 2, 3 and 4 refer respectively to the third, fourth, andfifth set of experiments.10

Consider the results in Table 1, which are based on the first set of experiments. Inthese experiments, the matrix of regressors in the null and alternative model are thesame and the only difference between the null and alternative pertains to the spatialweighting matrices. We will focus first on the size of the test, and then move to thepower.

Let us concentrate first on the top-left part of Table 1, where R2 = 0.35 and N =49. Looking at column averages, the empirical size of the test is quite close to thetheoretical 5% level only for the predictor y(2). In general, there are many cases inwhich the empirical size of the test exceeds the theoretical level. However, the “sizeof test problem” highlighted in the top part of the table mitigates when the value ofthe R2 = 0.68 (bottom-left part of the table). In this case, the empirical size of thetest based on both specifications of the error term, is, on average, quite close to thetheoretical 5% level. There are only few values outside of the acceptance interval. Inparticular, when the distribution of the errors is normal, there are only three cases andall of them are related to the predictor y(2). This is quite impressive given that thesample size is very small.

10We do not show any evidence for the sixth experiments. Of course, the results are available fromthe authors.

9

When the sample size increases but the value of R2 = 0.35 (as in the top-right partof Table 1), the empirical size of the test is still close to the theoretical 5% level only forthe predictor y(2). However, the average size for the predictor y(1) is much closer to theacceptance interval (.0611). Also there are less cases in the table where the estimate ofthe size is significantly different from the theoretical level.Fortunately, if R2 increases as well, the estimate of the size is always close to thetheoretical 5% level as it is revealed by the figures on the bottom-right part of thetable. In fact, there is not a single case in which the estimate of the size are significantlydifferent from the 5%.

Consider now the power estimates in Tables 1. Note that in all cases considered,consistently with the theoretical development, the power increases on average both withsample size and with the value of the R2. For example, the average power of the testwhen n = 49 and R2 = 0.35 (top-left) is .2370 (for the first predictor) and .2904 (forthe second predictor). The same average increases to .5623 (for the first predictor) and.5692 (for the second predictor) when the sample size is still n = 49 but R2 = 0.68(bottom-left). Additionally, when the sample size n = 100 and R2 = 0.35 (top-right)the average power of the test for the two predictors is .3836 and .4335. Finally, whenthe sample size n = 100 and R2 = 0.68 (bottom-right), the power of the test is close to75% with both predictors.

Of course, the power of the test also depends on the combination of ρ and λ. Specif-ically, for small values of λ (e.g. -0.2, 0.2) the power is always consistently lower thanin other cases. As an example, when n = 100 and R2 = 0.68 the value of the power forsmall values of λ ranges between .28 and .50. When λ is large, the value of the powerranges between .81 and 1.00.

Finally, it should be also noted that the power of the test corresponding to the useof y(1) is quite close, on average, to the power corresponding to the use of y(2).

Tables 2 is based on the third set of experiments. The structure of the table isidentical to that of the previous Table 1. In these experiments the null and alternativemodels differ in their regressor matrices while the spatial weighting matrix is the samefor both the null and alternative model.

Concentrating on Table 2, it is possible to note that only the predictor y(2), onaverage, has the expected size. For the predictor y(1) when N = 49 (left part of thetable), there are many cases in which the empirical size of the test exceeds the theoreticallevel (with both values of the R2). Interestingly, many of these cases are related to thepositive value of ρ = 0.4. When the sample size increases (right part), both y(1) andy(2) have, on average, the expected size even if the R2 is only .35. When the differencebetween the null and alternative model pertains to the matrix of regressors, our J-testpresents very high power even for relatively small sample size.

Table 3 is based on the fourth set of experiments and the sample size is N = 49.The null and alternative models differ in terms of both the matrix of regressors andthe spatial weighting matrix. This set of experiments is designed in such a way thatthe error term of the model under the alternative follows the same specification of the

10

error term under the null. Aside from parameter specifications, each section of Table3 is made up of four columns. The results in the first two columns (labeled “2sls”)are obtained by estimating the alternative model with a simple two stage least squareprocedure which ignores the details of the error specification- e.g., spatial correlation.The last two columns (labeled “2sls + gm”) report results obtained when the alternativemodel is estimated in the same way as the null: a procedure which accounts for spatialcorrelation. Both methods of estimation are consistent, but clearly the “2sls + gm”procedure is more efficient than the simple “2sls”.

The column averages of the empirical size of the test for both values of R2 andestimation procedures are pretty far away from the theoretical 5% level.11

Moving to the power of the test, we note from the table that, for all combinations ofmodel parameters and estimation procedures considered, the power of the test is almostalways equal to 1.0. In summary, the results in Table 3 suggest that there is almostno loss of power in the J-test if the alternative model is estimated by an inefficientprocedure as long as that procedure is consistent. Consistent with the results in Table2, the results in Table 3 also suggest that if the null and alternative models differ interms of their regressor matrices, the power of the J-test will be high.

The last two sets of experiments are specified in such a way that the null model istested against two possible alternatives. In what follows, we only present the fifth setof experiments (Table 4), where the models under the alternative differ from the nullonly in terms of the spatial weighting matrix.

Looking at the column averages in Table 5, all of the empirical size of the test areclose to the theoretical .05 level. However, the results corresponding to the individualsize estimates are not always close to the .05 level. Comparing these results with theresults in Table 1, while there seems to be an improvement in term of size, the valuesfor the power of the test are lower when the alternative specification includes more thanone possible model.

Summarizing our results we can conclude that the size of the test is not always closeto the nominal level when either the sample size is small or the error variance is large.The power of the test increases with sample size and there is not much difference if thealternative model is estimated efficiently other than consistently. However, the powerof the test is lower when the alternative specification includes more than one possiblealternative.

6 Empirical Application

The empirical application in this section is based on a well known economic modelof crime estimated by Cornwell and Trumbull (1994). They use a panel data on 90

11However, when the sample size increases (N = 100) the problem improves, and the empirical sizeof the test is on average close to the nominal value when R2 = 0.68. These results are available fromthe authors.

11

counties in North Carolina over the period 1981− 1987.12

The empirical model relates the crime rate13 to a series of variables which controlfor the return to legal opportunities, as well as a set of deterrent variables (such asprobability of arrest, probability of conviction conditional on arrest, and probability ofimprisonment conditional on conviction).

The ratio of arrests to offenses is a proxy for the probability of arrest; the ratioof convictions to arrest is a proxy for the probability of conviction, and, finally, theproportion of total convictions resulting in prison sentences is a proxy for the probabilityof imprisonment. The model also includes a measure of sanction severity measuredby the average prison sentence length in days. All of the other variables are eitherobservable county characteristics, or controls for the relative return to legal activities.14

Cornwell and Trumbull (1994) estimate the model both by the between and thewithin estimators. In their estimation, they account for the endogeneity of the policeper-capita and the probability of arrest variables. They use two instruments for thesevariables, namely an offense mix variable (the ratio of crimes involving face to facecontacts to those that do not) and per capita tax revenue.15

As suggested by Brueckner (2003), a crime model is a strategic interactions model,and, in particular, a spillover model. In this framework each county “chooses” the levelof crime by choosing the level of enforcement, e.g., the extent of the police force. Atthe same time, however, the county will be affected by the decision process made byother counties, thus indicating the presence of spillover.16

The aim of our empirical application is twofold. First of all, we want to takea spatial explicit approach in estimating the crime model of Cornwell and Trumbull(1994). Additionally we want to test the hypothesis that the spatial weights matrixshould be based on a contiguity criteria and not on a distance based approach. Theexplanation behind this second hypothesis is that county i may only be concerned ofthe decisions that immediate neighbors are taking, and ignore counties that are moredistant.

With our J-test we wish to test two competing non-nested alternatives. The nullmodel is a modification of the one estimated by Cornwell and Trumbull (1994) thatincludes a spatial lag of the dependent variable (i.e., the spatial lag of the crime rate)as well as spatial correlation in the errors. This null model is specified in terms of aspatial weights matrix based on the ten nearest neighbors. On the other hand, thealternative model is identical to the null except that the spatial weights matrix is

12This dataset is well known because it is one of the datasets in Baltagi (2008). The data areavailable from the website associated with Baltagi’s book.

13The crime rate variable is the ratio between an FBI index that measures the number of crimes,and county population (i.e. crime per-capita in the county).

14For greater details on the data see Baltagi (2008).15See Baltagi (2008) for further explanation on the instruments.16In other terms we are estimating a reaction function which gives county i’s best reaction to the

choices of other counties.

12

specified in terms of contiguity.17

As indicated, the J-test is based on augmenting the null model by predictions basedon the alternative model. The procedure then is to test for the significance of theaugmenting variable.

At the 5% level the J-test rejects the null model since the Chi-squared variable= 6.373 > χ2

1 = 3.841.18 We conclude then that counties only consider policies im-plemented in boarder counties and do not consider those of counties that are moredistant.

7 Conclusions

In the present paper we have extended the J-test to a panel data framework. OurJ-test is suitable for testing a null spatial model against one or more alternative spatialmodels. These alternative models may differ from the null either in their regressormatrix, their weighting matrix, or both. The J-test is not appropriate for testing analternative model which only differs from the null in its error specifications.

Our suggested test is computationally simple. Its size in relatively small panel datasamples is reasonably close to the theoretical size, except when the sample size is verysmall. On the other hand, the power of the test is high, and seems to be only mildlyaffected by the number of models in the alternative.

A suggestion for future research would be an extension of our results to panel datamodels which has fixed effects, as well as both spatially and time lagged dependentvariables. Another suggestion for future research would be an extension to a non-linearspatial panel framework. Among others, such a framework would arise in a qualitativeor a limited dependent variable setting.

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16

Table 1: Frequency of rejection of the Null Hypothesis: two predictors (2,000 replica-tions).

First set of experiments:H0: X0, W1, H1: X0, W2

εn = N(0, In) εn = N(0, In)n = 49 n = 100

Size Power Size Power

R2 = 0.35

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0535 0.0430 0.2830 0.1825 0.0555 0.0515 0.5660 0.5115λ=-0.4 0.0580 0.0425 0.1860 0.0745 0.0580 0.0460 0.3230 0.2435λ=-0.2 0.0650 0.0460 0.0990 0.0335 0.0605 0.0460 0.1410 0.0815λ=0.2 0.0600 0.0525 0.0955 0.1880 0.0555 0.0535 0.1335 0.2400λ=0.4 0.0790 0.0495 0.2450 0.4670 0.0615 0.0525 0.4025 0.6215λ=0.6 0.0850 0.0470 0.4655 0.7700 0.0625 0.0485 0.7275 0.9210

ρ=0.4 λ=-0.6 0.0665 0.0900 0.3155 0.1890 0.0565 0.0590 0.5820 0.5075λ=-0.4 0.0720 0.0890 0.2065 0.0955 0.0560 0.0545 0.3445 0.2580λ=-0.2 0.0660 0.0805 0.1125 0.0460 0.0490 0.0500 0.1185 0.0700λ=0.2 0.0810 0.0660 0.1020 0.1785 0.0610 0.0505 0.1255 0.2110λ=0.4 0.0925 0.0575 0.2420 0.4755 0.0760 0.0470 0.4050 0.6140λ=0.6 0.1000 0.0465 0.4920 0.7845 0.0815 0.0440 0.7345 0.9225

Averages 0.0732 0.0592 0.2370 0.2904 0.0611 0.0503 0.3836 0.4335

R2 = 0.68

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0530 0.0555 0.8035 0.7500 0.0510 0.0515 0.9905 0.9900λ=-0.4 0.0555 0.0590 0.4985 0.4040 0.0535 0.0490 0.8465 0.8185λ=-0.2 0.0565 0.0535 0.1765 0.1165 0.0575 0.0470 0.3605 0.3100λ=0.2 0.0540 0.0540 0.2300 0.3175 0.0510 0.0550 0.4120 0.5050λ=0.4 0.0575 0.0540 0.7100 0.8205 0.0590 0.0560 0.9395 0.9700λ=0.6 0.0580 0.0445 0.9570 0.9870 0.0540 0.0510 0.9995 1.0000

ρ=0.4 λ=-0.6 0.0510 0.0655 0.7985 0.7460 0.0535 0.0560 0.9940 0.9910λ=-0.4 0.0505 0.0635 0.4935 0.4215 0.0440 0.0520 0.8680 0.8370λ=-0.2 0.0525 0.0695 0.1815 0.1315 0.0430 0.0480 0.3450 0.2855λ=0.2 0.0460 0.0525 0.2305 0.3215 0.0455 0.0495 0.3960 0.4930λ=0.4 0.0510 0.0590 0.7100 0.8270 0.0535 0.0505 0.9305 0.9685λ=0.6 0.0520 0.0440 0.9580 0.9875 0.0540 0.0515 0.9990 1.0000

Averages 0.0531 0.0562 0.5623 0.5692 0.0516 0.0514 0.7568 0.7640

17

Table 2: Frequency of rejection of the Null Hypothesis: two predictors (2,000 replica-tions).

Third set of experiments:H0: X0, W1, H1: X1, W1

εn = N(0, In) εn = N(0, In)n = 49 n = 100

Size Power Size Power

R2 = 0.35

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0600 0.0510 1.0000 1.0000 0.0645 0.0555 1.0000 1.0000λ=-0.4 0.0640 0.0515 1.0000 1.0000 0.0630 0.0565 1.0000 1.0000λ=-0.2 0.0440 0.0340 1.0000 1.0000 0.0545 0.0500 1.0000 1.0000λ=0.2 0.0435 0.0495 1.0000 1.0000 0.0440 0.0505 1.0000 1.0000λ=0.4 0.0510 0.0510 0.9995 1.0000 0.0460 0.0535 1.0000 1.0000λ=0.6 0.0470 0.0455 0.9990 1.0000 0.0365 0.0540 1.0000 1.0000

ρ=0.4 λ=-0.6 0.0800 0.0570 1.0000 1.0000 0.0620 0.0475 1.0000 1.0000λ=-0.4 0.0895 0.0600 1.0000 1.0000 0.0600 0.0545 1.0000 1.0000λ=-0.2 0.0760 0.0550 1.0000 1.0000 0.0600 0.0585 1.0000 1.0000λ=0.2 0.0695 0.0490 1.0000 1.0000 0.0545 0.0540 1.0000 1.0000λ=0.4 0.0770 0.0525 1.0000 1.0000 0.0500 0.0525 0.9995 1.0000λ=0.6 0.0625 0.0470 0.9980 1.0000 0.0440 0.0545 0.9995 1.0000

Averages 0.0637 0.0503 0.9997 1.0000 0.0533 0.0535 0.9999 1.0000

R2 = 0.68

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0525 0.0510 1.0000 1.0000 0.0610 0.0575 1.0000 1.0000λ=-0.4 0.0570 0.0545 1.0000 1.0000 0.0605 0.0580 1.0000 1.0000λ=-0.2 0.0445 0.0380 1.0000 1.0000 0.0505 0.0510 1.0000 1.0000λ=0.2 0.0520 0.0510 1.0000 1.0000 0.0465 0.0460 1.0000 1.0000λ=0.4 0.0605 0.0550 1.0000 1.0000 0.0500 0.0515 1.0000 1.0000λ=0.6 0.0555 0.0550 1.0000 1.0000 0.0410 0.0465 1.0000 1.0000

ρ=0.4 λ=-0.6 0.0680 0.0595 1.0000 1.0000 0.0455 0.0470 1.0000 1.0000λ=-0.4 0.0770 0.0675 1.0000 1.0000 0.0490 0.0485 1.0000 1.0000λ=-0.2 0.0665 0.0530 1.0000 1.0000 0.0580 0.0560 1.0000 1.0000λ=0.2 0.0705 0.0595 1.0000 1.0000 0.0475 0.0515 1.0000 1.0000λ=0.4 0.0700 0.0545 1.0000 1.0000 0.0450 0.0490 1.0000 1.0000λ=0.6 0.0635 0.0535 1.0000 1.0000 0.0400 0.0450 1.0000 1.0000

Averages 0.0615 0.0543 1.0000 1.0000 0.0495 0.0506 1.0000 1.0000

18

Table 3: Frequency of rejection of the Null Hypothesis: two predictors (2,000 replica-tions).

Fourth set of experiments:H0: X0, W1, H1: X1, W3

εn = N(0, In)Size Size Power Power2sls 2sls + gm 2sls 2sls + gm

N = 49, R2 = 0.35

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0680 0.0840 0.0780 0.0740 0.9965 0.9980 1.0000 1.0000λ=-0.4 0.0815 0.1000 0.0880 0.0925 0.9995 0.9995 1.0000 1.0000λ=-0.2 0.0730 0.0895 0.0765 0.0795 1.0000 1.0000 1.0000 1.0000λ=0.2 0.0860 0.1005 0.0855 0.0890 1.0000 1.0000 1.0000 1.0000λ=0.4 0.0705 0.0925 0.0790 0.0870 1.0000 1.0000 1.0000 1.0000λ=0.6 0.0620 0.0700 0.0675 0.0790 1.0000 1.0000 1.0000 1.0000

ρ=0.4 λ=-0.6 0.0730 0.0975 0.0775 0.0805 1.0000 1.0000 1.0000 1.0000λ=-0.4 0.0900 0.1150 0.0950 0.0995 1.0000 1.0000 1.0000 1.0000λ=-0.2 0.0740 0.1005 0.0850 0.0950 1.0000 1.0000 1.0000 1.0000λ=0.2 0.0705 0.0935 0.0735 0.0805 1.0000 1.0000 1.0000 1.0000λ=0.4 0.0615 0.0755 0.0730 0.0810 0.9985 0.9975 1.0000 1.0000λ=0.6 0.0565 0.0670 0.0670 0.0745 0.9890 0.9885 1.0000 1.0000

Averages 0.0722 0.0905 0.0788 0.0843 0.9986 0.9986 1.0000 1.0000

N = 49, R2 = 0.68

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0560 0.0630 0.0545 0.0560 1.0000 1.0000 1.0000 1.0000λ=-0.4 0.0695 0.0755 0.0710 0.0745 1.0000 1.0000 1.0000 1.0000λ=-0.2 0.0585 0.0610 0.0585 0.0545 1.0000 1.0000 1.0000 1.0000λ=0.2 0.0675 0.0755 0.0645 0.0720 1.0000 1.0000 1.0000 1.0000λ=0.4 0.0605 0.0645 0.0660 0.0685 1.0000 1.0000 1.0000 1.0000λ=0.6 0.0555 0.0555 0.0610 0.0605 1.0000 1.0000 1.0000 1.0000

ρ=0.4 λ=-0.6 0.0610 0.0650 0.0595 0.0610 1.0000 1.0000 1.0000 1.0000λ=-0.4 0.0745 0.0820 0.0755 0.0750 1.0000 1.0000 1.0000 1.0000λ=-0.2 0.0685 0.0735 0.0665 0.0690 1.0000 1.0000 1.0000 1.0000λ=0.2 0.0560 0.0640 0.0585 0.0670 1.0000 1.0000 1.0000 1.0000λ=0.4 0.0540 0.0595 0.0620 0.0625 1.0000 1.0000 1.0000 1.0000λ=0.6 0.0515 0.0580 0.0580 0.0640 1.0000 1.0000 1.0000 1.0000

Averages 0.0611 0.0664 0.0630 0.0654 1.0000 1.0000 1.0000 1.0000

19

Table 4: Frequency of rejection of the Null Hypothesis: two predictors (2,000 replica-tions).

Fifth set of experiments:H0: X0, W3, H1: X0, W1, W2

εn = N(0, In) εn = N(0, In)n = 49 n = 100

Size Power Size Power

R2 = 0.35

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0515 0.0465 0.1315 0.1650 0.0455 0.0455 0.3865 0.4195λ=-0.4 0.0535 0.0420 0.0860 0.0820 0.0490 0.0495 0.1930 0.1705λ=-0.2 0.0500 0.0525 0.0490 0.0315 0.0580 0.0565 0.0920 0.0760λ=0.2 0.0500 0.0425 0.0610 0.0990 0.0665 0.0560 0.1195 0.1670λ=0.4 0.0605 0.0485 0.1430 0.3150 0.0725 0.0495 0.3745 0.5465λ=0.6 0.0695 0.0405 0.2720 0.6300 0.0940 0.0435 0.7335 0.9155

ρ=0.4 λ=-0.6 0.0620 0.0760 0.1250 0.1590 0.0545 0.0590 0.3940 0.4065λ=-0.4 0.0530 0.0840 0.0810 0.0780 0.0490 0.0595 0.2060 0.1805λ=-0.2 0.0585 0.0825 0.0500 0.0365 0.0505 0.0675 0.0825 0.0595λ=0.2 0.0505 0.0450 0.0630 0.1090 0.0515 0.0405 0.1095 0.1645λ=0.4 0.0605 0.0340 0.1485 0.3125 0.0540 0.0345 0.3890 0.5490λ=0.6 0.0620 0.0210 0.2880 0.6670 0.0670 0.0300 0.7345 0.9075

Averages 0.0568 0.0513 0.1248 0.2237 0.0593 0.0493 0.3179 0.3802

R2 = 0.68

y(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i y

(1)i y

(2)i

ρ=-0.4 λ=-0.6 0.0505 0.0455 0.5655 0.6095 0.0505 0.0500 0.9675 0.9675λ=-0.4 0.0580 0.0520 0.3055 0.2885 0.0505 0.0530 0.7200 0.7085λ=-0.2 0.0570 0.0535 0.1020 0.0870 0.0595 0.0575 0.2335 0.2255λ=0.2 0.0520 0.0510 0.1575 0.2015 0.0635 0.0565 0.3480 0.4170λ=0.4 0.0685 0.0555 0.4965 0.6390 0.0605 0.0475 0.9285 0.9615λ=0.6 0.0750 0.0430 0.8540 0.9480 0.0750 0.0535 0.9995 1.0000

ρ=0.4 λ=-0.6 0.0545 0.0585 0.5375 0.5800 0.0535 0.0515 0.9465 0.9505λ=-0.4 0.0585 0.0710 0.2925 0.2990 0.0545 0.0565 0.7055 0.6945λ=-0.2 0.0590 0.0750 0.0975 0.0900 0.0565 0.0595 0.2380 0.2135λ=0.2 0.0470 0.0555 0.1405 0.1960 0.0455 0.0510 0.3580 0.4300λ=0.4 0.0565 0.0515 0.4990 0.6490 0.0500 0.0450 0.9315 0.9615λ=0.6 0.0710 0.0310 0.8810 0.9625 0.0610 0.0415 0.9995 1.0000

Averages 0.0590 0.0536 0.4108 0.4625 0.0567 0.0519 0.6980 0.7108

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Appendix

The VC matrix of ηJ : The model for yi,t under H1,J in (5) is

yi,t = Pi,t,JβJ + λJMi,Jyt + Yi,t,JdJ + µJ + ψi,t,J (A.1)

where Pi,t,J and Mi,J are, respectively, the ith rows of Pt,J and MJ , and ψi,t,J is the ith

element of ψt,J . The mean of yi,t given Φ2,i,t,J = [Φ1,J ,Mi,Jyt, Yi,t,J ] is

E(yi,t|H1,J ,Φ2,i,t,J) = Pi,t,JβJ + λJMi,Jyt + Yi,t,JdJ + µi,J + ηi,t,J (A.2)

where ηi,t,J = E(ψi,t|H1,J ,Φ2,i,t,J) 6= 0, because of the correlation of Mi,Jyt and Yi,t,Jwith ψi,t,J . However, using iterated expectations,

E(ηi,t,J |H1,J ,Φ1,J) = E [E(ψi,t,J |H1,J ,Φ2,i,t,J)|H1,J ,Φ1,J ] (23)

= E [E(ψi,t,J |H1,J ,Φ1,J ,Mi,Jyt, Yi,t,J)|H1,J ,Φ1,J ]

= E(ψi,t,J |H1,J ,Φ1,J) = 0

Since ηi,t,J is the i, tth element of ηJ , it follows from (A.3) that E(ηJ |H1,J ,Φ1,J) =0. Since E(ψi,t,J |H1,J ,Φ2,i,t,J) = ηi,t,J , given H1,J , ηi,t,J is a function of Φ2,i,t,J , sayf(Φ2,i,t,J), since a conditional mean is a function of the conditioning variables. It alsofollows that ψi,t,J can be expressed as

ψi,t,J = ηi,t,J + ϕi,t,J ; E(ϕi,t,J |H1,J ,Φ2,i,t,J) = 0 (A.4)

and soE(ϕi,t,J ηi,t,J |H1,J ,Φ2,i,t,J) = ηi,t,JE(ϕi,t,J |H1,J ,Φ2,i,t,J) = 0 (A.5)

Again, via iterated expectations,

E(ϕi,t,J ηi,t,J |H1,J ,Φ1,J) = 0 (A.6)

Using evident notation, let ϕJ and ηJ be the NT × 1 vectors of ϕi,t,J and ηi,t,J , i =1, ..., N ; t = 1, ..., T ; J = 1, ..., G. Then, from (8), and (A.3) - (A.6)

E(ψJψ′J |H1,J ,Φ1,J) = ΩψJ

= ΩϕJ+ ΩηJ

where ΩϕJand ΩηJ are the VC matrices, respectively, of ϕJ and ηJ . Since the row

and column sums of ΩψJare uniformly bounded in absolute value so are the row and

column sums of ΩϕJand ΩηJ .

The asymptotic negligibility of ηJ when G=1 : Since Q0(eT ⊗ µJ) = 0, thesecond line of (10) follows from the stacked version of (A.2). If the researcher observed

21

ηJ and so took used Q0ηj as part of the predictor Q0E(y|H1,J ,Φ2,J) in (10), the only

term involving Q0ηj in (NT )1/2[δ − δ] based on (16) and (17) would be ∆J where

∆J = (NT )−1[H ′(INT − ρ0(IT ⊗W )]Q0ηJ= (NT )−1H ′Q0ηJ − ρ0(NT )−1[H ′(IT ⊗W )]Q0ηJ

Given the results in Mutl and Pfaffermayr (2011) and Kapoor et al. (2007), ρ0P→ ρ0.

Assume as in Mutl and Pfaffermayr (2011) and Kapoor et al. (2007), that the row andcolumn sums of W are uniformly bound in absolute value, that the elements of H areuniformly bounded in absolute value, and (NT )−1H ′H → QHH where QHH is finiteand nonsingular.

Let FJ,1 = (NT )−1H ′Q0ηJ and FJ,2 = (NT )−1[H ′(IT ⊗ W )]Q0ηJ so that ∆J =FJ,1 + ρ0FJ,2. From (A.3) we have E(FJ,q|H1,J ,Φ1,J) = 0, q = 1, 2. Also, the VCmatrices of FJ,1 and FJ,2 are, respectively, ΩFJ,1

= (NT )−2H ′Q0ΩηJQ0H and ΩFJ,2=

(NT )−2[H ′(IT ⊗W )]Q0ΩηJQ0(IT ⊗W ′)H]. Since the row and column sums of W,Q0

and ΩηJ are uniformly bounded in absolute value, and the elements of H are uniformly

bounded, the elements of ΩFJ,1and ΩFJ,1

are all 0(N−1)→ 0. Since ρ0P→ ρ0, ΩFJ,q

P→ 0

and so, by Chebyshev’s inequality, ∆JP→ 0.

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