Specification Testing for Panel Spatial Models with Misspecifications

Transcription

1 Specification Testing for Panel Spatial Models with Misspecifications Monalisa Sen and Anil K. Bera University of Illinois at Urbana Champaign (Working paper-not complete) January Abstract Specification of a model is one of the most fundamental problems in econometrics. In most cases, specification tests are carried out in a piecemeal fashion, for example, testing the presence of one-effect at a time ignoring the potential presence of one-effect at a time ignoring the potential presence of other forms of misspecification. It is also much to expect from a practioners to estimate a very complex model and then to carry out specification tests. Specification problems are paramount in the spatial panel models which are now increasingly being used in practice given the wide availability of longitudinal data. Using Bera and Yoon (993) general test procedure under misspecification, in the context of simple spatial model, Anselin, Bera, Florax, Yoon (996) developed ordinary least squares (OLS) based robust Rao s Score (RS) test for lag(error) dependence in the possible presence of error (lag) dependence. In similar fashion, for panel data model, Bera, Sosa-Escudero, Yoon () developed adjusted RS tests (for random effect and serial correlation) that can identify the correct source(s) of misspecification. In the context of specification tests for spatial panel models, there had been very important contributions, for example, see Baltagi, Song and Koh (3). Baltagi et. al. (7), Baltagi and Liu (8) and Montes-Rojas (). Many of the suggested tests require estimation of complex models and even then these tests cannot take account of multiple forms of departures, such as: the spatial lag, spatial error, random effect and time-series serial correlation. Using our general model we first propose an overall test for all possible misspecifications. Then we derive a number of adjusted RS tests that can identify the definite cause(s) of rejection of the basic model and thus adding in the steps for model revision. For empirical researchers, our suggested procedures provide simple strategies for model testing using OLS residuals from standard linear model for spatial panel data. Through an extensive simulation study, we evaluate the finite sample performance of our suggested tests and available procedures. We also formulate a step-by-step sequential strategy for use in empirical practice. Finally, to illustrate the usefulness of our procedures, we provide some empirical applications in the context of the convergence theory of incomes of different economies which is a widely studied empirical problem in macro-economic growth theory.

2 . Introduction Econometricians interest on problems that arise when the assumed model (used in constructing a test) deviates from the data generating process (DGP) goes a long way back. As emphasized by Haavelmo (944), in testing any economic relations, specification of a set of possible alternatives, called the priori admissible hypothesis,ω, is of fundamental importance. Misspecification of the priori admissible hypotheses was termed as type-iii error by Bera and Yoon (993), and Welsh (996, p. 9) also pointed out a similar concept in the statistics literature. Broadly speaking, the alternative hypothesis may be misspecified in three different ways. In the first one, what we shall call complete misspecification, the set of assumed alternatives, Ω, and the DGP Ω,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 occurs when the alternative is underspecified in that it is a subset of a more general model representing the DGP, i.e., Ω Ω. This happens, for example, when both serial correlation and individual effects are present, but are tested separately (one at a time). The last case is overtesting which results from overspecification, i.e., when Ω Ω.This can happen if a joint test for serial correlation and random individual effects is conducted when only one effect is present in DGP. It can be expected that consequences of overtesting will 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 (98) and Bera ()] Using the asymptotic distributions of standard Rao s Score(RS) test under local misspecification, Bera and Yoon(993) suggested an adjusted RS test that is robust under misspecification and asymptotically equivalent to the optimal Neyman s C(α) test. As we will discuss, an attractive

3 feature of this approach is that the adjusted test is based on the joint null hypothesis of no misspecifications, thereby requiring estimation of the simplest model. A surprising additivity property also enables us to calculate the adjusted tests quite effortlessly. The origins of specification testing in spatial econometrics can be traced back to Moran (95a, 95b). Much later this area was further enriched by many researchers, for example, see Cliff and Ord(97), Brandsma and Ketellapa(979a, 979b), Burridge (98), Anselin (98,988a,988b,988c) and Kelejian and Robinson (99). 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. As we discussed above, their separate applications 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 autocorrelations. Apart from the problem of overtesting (when only one kind of dependence characterizes the DGP) mentioned above, the problem with such a test is that we cannot identify the exact nature of spatial dependence when the joint null hypothesis is rejected. One approach to deal with this problem to use conditional tests, i.e., to use test for spatial error dependence after estimating a spatial lag model, and vice versa. This, however, requires maximum likelihood (ML) estimation, and the simplicity of test based on OLS residuals is lost. Anselin, Bera, Florax and Yoon (996) 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 (993) procedure, Anselin et.al (996) 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 good finite sample size and power properties. 3

4 In a similar fashion, in context of panel data model, Bera Sosa-Escudero and Yoon () showed that when one tests for either random effects or serial correlation without taking account of the presence of other effect, the test have reject the true null hypothesis far too often under the presence of the unconsidered parameter. In particular, they found that the presence of serial correlation made the Bruesh and Pagan (98) test for random effects to have excessive size. Similar over rejection occur for the test of serial correlation when the presence of random effect is ignored. Bera et. al () developed a 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 (996) and Bera et. al (), we have the spatial panel model, potentially with four sources of departure (from the classical regression model) coming from four extra parameters: the spatial lag, spatial error, random effect and(time series) serial correlation parameters. In this paper, we investigate a number of strategies to test against multiple form of misspecification of this kind. Using our general model we derive an overall test and a number of adjusted tests that take the account of possible misspecifications 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. As we further discuss below, all the available tests in the literature take into account of only two potential sources of misspecifications at a time, and many of them require ML estimation to account for the nuisance parameter(s). Recently, many researchers have conducted conditional and marginal specification tests in spatial panel data models. Baltagi, Song and Koh (3) proposed conditional LM tests, which test for random regional effects given the presence of spatial error correlation and also, spatial 4

5 error correlation given the presence of random regional effects. Baltagi et al (7) adds 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 (8) also developed similar LM and LR tests with spatial lag dependence and random individual effects in a panel data regression model. Their paper derives two conditional LM tests; first one test for the absence of random individual effects without ignoring the possible presence of spatial lag dependence and the second one test for the absence of spatial lag dependence without ignoring the possible presence of random individual effects. Baltagi, Song and Kwon (9) considered a panel data regression with heteroscedasticity as well as spatially correlated disturbances. Like the previous works, Baltagi et al (9) derived the conditional LM and marginal LM tests. But all the specification tests proposed as in Baltagi, Song and Koh (3), Baltagi et al (7), Baltagi and Liu(8), Baltagi et al (9) require ML estimation of individual effects parameters. One directional marginal tests and conditional tests get more and more complicated as we add more and more parameters to generalize the model in multiple directions. Recently, Gabriel Montes-Rojas () has proposed an adjusted LM test (based on Bera and Yoon (993)) for spatial autoregressive (SAR) panel data model with autocorrelation of errors and random effects. In his paper he has proposed the test of autocorrelation (regional random effects) in presence of regional random effects (autocorrelation). He derived the locally adjusted RS test of one parameter in presence of one nuisance parameter; thereby estimating the spatial dependent parameter using MLE and IV estimation strategy. Section gives the main results on the distribution of Rao Score (RS) tests when the alternative hypothesis is misspecified, and presents the modified RS test which is robust under local misspecification. We develop the spatial panel model framework in Section 3 and derive the 5

6 log-likelihood functions. Section 4 derives the new diagnostic tests which take account of misspecifications in multiple directions. In Section 5 we provide evidence on the performance of robust tests based on Monte Carlo simulations. Section 6 gives a brief background of the different kinds of empirical growth theory (income-convergence) models and its problems and illustrates how our robust specification tests can avoid misspecification errors. In particular, we use Heston, Summers and Aten() Penn World Table, which contains information on real income, investment and population(among many other variables) for a large number of countries and the growth-model proposed by Ertur and Koch (7) to illustrate the usefulness of our proposed tests. Section 7 concludes the paper.. A General Approach to Testing in the Presence of Nuisance Parameters Consider a general model represented by the log-likelihood L(γ, ψ, φ) where the parameters γ, ψ and φ are respectively (p x ), (r x ) and (s x ) vectors. Suppose a researcher sets φ = φ and tests H : ψ = ψ using the log-likelihood function L (γ, ψ) = L(γ, ψ, φ ), where ψ and φ are known. The Rao Score statistic for testing H in L (γ, ψ) will be denoted by RS ψ. Let us denote θ = (γ, ψ, φ ) (p+r+s) and θ = (γ, ψ, φ ), where γ is the ML estimator of γ when (p+r+s) ψ = ψ and φ = φ. We define the score vector and the information matrix, respectively, as d a (θ) = L(θ) a for a= (γ, ψ, φ), and J(θ) = E n L(θ) θ θ J γp J x p γψp J x r γφp x s = J ψγr J x p ψr J x r ψφr x s J φγs J x p φψs J x r φs x s (p+r+s)x(p+r+s) () 6

7 where n denotes the sample size. If L (γ, ψ) were the true model, then it is well known that under H : ψ = ψ, RS ψ = d ψ θ J ψ.γ θd ψ θ D χ r (), () where D denotes convergence in distribution and J ψ.γ (θ) J ψ.γ = J ψ J ψγ J γ J γψ. (3) Under local alternatives H : ψ = ψ + ξ n, RS ψ D χ r (λ ), where the non centrality parameter λ λ (ξ) = ξ J ψ.γ ξ. Given this set-up, asymptotically the test will have correct size and will be locally optimal [See Bera and Bilias JSPI ()]. Now suppose that the true log-likelihood function is L (γ, φ) so that the considered alternative L (γ, ψ) is (completely) misspecified. Using the local misspecification φ = φ + δ/ n, Davidson and MacKinnon (987) and Saikkonen (989) derived the asymptotic distribution of RS ψ under L (γ, φ) as RS ψ D χr (λ ), where the non-centrality parameter λ λ (δ) = δ J φψ.γ J ψ.γ J ψφ.γ δ with J ψφ.γ = J ψφrxs J ψγrxp J γpxp J γφpxs. (4) Owing to the presence of this non-centrality parameter λ, RS ψ will reject the null hypothesis H : ψ = ψ more often than allowed by the size of the test. Even when ψ = ψ is true, this leads the test to have excessive size. Therefore, the test will have incorrect size. Here the crucial term is J ψφ.γ (Eq. 4) which can be interpreted as the partial covariance between the score vectors d ψ and d φ after eliminating the linear effect of d γ on d ψ and d φ. If J ψφ.γ =, then asymptotically the local presence of the parameter φ has no effect on RS ψ. Using (4), Bera and Yoon (993) suggested a modification to RS ψ so that the resulting test is valid in the local presence of φ. The modified statistic is given by RS ψ = d ψ θ J ψφ.γ θj φ.γ θ d φ θ J ψ.γ θ J ψφ.γ θj φ.γ θ J φψ.γ θ d ψ θ J ψφ.γ θj φ.γ θ d φ θ (5) 7

8 This new test essentially adjusts the mean and variance of the standard RS ψ. Another way to look at RS ψ is to view the quantity (J ψφ.γ θj φ.γ θ d φ θ ) as the prediction of d ψ θ by d φ θ, and thus (J ψφ.γ θj φ.γ θ d φ θ ) = d ψ.φ θ say, is the part of d ψ θ that remains after eliminating the effect of d φ θ. In the literature, d ψ.φ θ is known as the effective score of ψ (that is orthogonal to d φ θ.[ see Bera and Bilias ()] Under ψ = ψ and φ = φ + δ/ n, RS ψ has a central χ r distribution. Thus, under misspecification RS ψ has the same asymptotic null distribution central χ r as that of RS ψ with ψ = ψ and φ = φ, thereby producing an asymptotically correct size test even when the model is locally misspecified. Bera and Yoon (993) further show that for local misspecification the adjusted test is asymptotically equivalent to Neyman s C(α) test and shares the optimal properties of the test. Three observations are worth noting regarding RS ψ. First, RS ψ requires estimation only under the joint null, namely ψ = ψ and φ = φ. That means, in most cases, we can conduct our tests based on only OLS residuals. Given the full specification of the model L(γ, ψ, φ), it is of course possible to derive RS test for ψ = ψ in the presence of φ which are generally referred to as conditional tests. However, that requires ML estimation of φ which could be difficult in some instances. Second, when J ψφ.γ =, RS ψ = RS ψ. In practice this is a simple condition to check, and as we mentioned before, if this condition is true, RS ψ (Eq. ) is an asymptotically valid test in the local presence of φ. Finally, let RS ψφ denote the joint RS test statistic for testing hypothesis of the form H : ψ = ψ and φ = φ using the alternative model L(γ, ψ, φ). Then it be shown that [Bera, Bilias and Yoon(7)] RS ψφ = RS ψ + RS φ = RS φ + RS ψ (6) 8

9 where RS φ and RS φ are respectively, the counterparts of RS ψ and RS ψ for testing H : φ = φ. This is a very important identity since it implies that a joint RS test for two parameter vectors ψ and φ can be decomposed into sum of two orthogonal components: the adjusted statistic for one parameter vector and (unadjusted) marginal test statistic for the other. Since many econometrics softwares provide the marginal (and sometime the joint) test statistics, the adjusted versions can be obtained effortlessly. In the context of spatial panel model ψ and φ will denote any combinations of the four parameters relating spatial lag, spatial error, random effect and serial correlation; and the parameter vector γ will correspond to basic regression model. All the test statistic in (6) are based on the OLS estimator γ under the joint null of absence of any kind of correlations and random effect, i.e., under joint H : ψ = ψ and φ = φ, say. In a way RS ψφ can be viewed as the total departure from the joint null hypothesis of no misspecification in the basic regression model. When the true model is presented by L(γ, ψ, φ), RS ψφ is always a valid test, and under local alternatives ψ = ψ + ξ/ n and φ = φ + δ/ n D RS ψφ χr+s (λ 3 ), where λ 3 = λ 3 (ξ, δ) = [ξ δ ] J ψ.γ J ψφ.γ ξ. [See Bera et. al()] J φ.γ δ J φψ.γ Significance of RS ψφ indicates some form of misspecification in the basic model (involving only parameter vector γ). However, we can identify the correct source(s) of departure only by using the adjusted statistics (RS ψ and RS φ ) not the marginal ones (RS φ and RS ψ ). Our testing strategy is close to the idea of Hillier (99) in the sense that we try to partition an overall rejection region to obtain evidence about the correct direction(s) in which the basic model needs revision. And we achieve that without estimating any of the nuisance parameters. 9

10 3. A Spatial Panel Model We consider the following spatial panel model: N y it = τ j= m ij y jt + X it β + u it, for i =,,, N; t =,,, T, (7) u it = μ i + ε it (8) ε it = λ N j= w ij ε jt + v it (9) v it = ρv it + e it, where e it ~IIDN(, σ e ). () Here y it is the observation for the i th location/individual at the t th time period, X it denotes the k x vector of observations on the nonstochastic regressors and u it is the regression disturbance. In addition to spatially dependence of y through the weight matrix M = ((m ij )), we have random effects, μ i, such that μ i ~IID(, σ µ ), spatially autocorrelated residual disturbances and a first order autoregressive remainder disturbance term. Here, τ and λ are respectively the spatial lag and spatial error dependence parameters with τ < and λ <, and ρ is the ( time-series) first-order correlation coefficient satisfying ρ <. W and M are known N x N spatial row-standardized weight matrix whose diagonal elements are zero, such that (I N τm) and (I N λw) are non-singular, where I N is an identity matrix of dimension N. Using the above equations (7-) we can express the model in compact notation as y = τ(i T M)y + Xβ + u, () where y is of dimension NT x, X is NT x k, β is k x and u is NT x. X is assumed to be of full column rank and its elements are bounded in absolute value. The disturbance term can be written as u = (ι T I N )μ + (I T B )v, ()

11 where B = (I N λw). ι T is a vector of ones of dimension T, I T is an identity matrix of dimension T and denotes the Kronecker product. Under our setup the variance covariance matrix of u is given by Ω = σ μ (J T I N ) + [V (B B) ] = σ e (J T I N )φ + V ρ (B B), (3) where δ = σ μ / σ e, J T is a matrix of ones of dimension T, and V is the familiar AR() variance covariance matrix of dimension T, V = E(v v) = σ e ρ V = σ e V ρ, (4) where ρ ρ ρ T V = and V ρ = ρ T ρ T ρ T 3 ρ V. The loglikelihood function of the above model can be written as follows: L = NT lnπ ln Ω + Tln A [(I T A)y Xβ] Ω [(I T A)y Xβ] (5) where A = (I N τm), and following Baltagi et. al (7) ln Ω = N ln( ρ ) + ln d ( ρ) φi N + (B B) + NT lnσ e (T ) ln B, with d = α + (T )and α = +ρ. Thus substituting ln Ω in L (Eq. 5) we obtain ρ L = NT lnπ N ln( ρ ) ln d ( ρ) φi N + (B B) NT lnσ e + (T ) ln B + Tln A [(I T A)y Xβ] Ω [(I T A)y Xβ]. (6)

12 4. Derivation of the Specification Tests We are interested in testing H : ψ = in the possible presence of the parameter vector φ. For the spatial panel model the full parameter vector is given by θ = (β, σ e, σ μ, ρ, λ, τ). Ιn context of our earlier notation θ = (γ, ψ, φ ), γ = ( β, σ e ) and ψ and φ could be any combinations of the parameters under test, namely (σ μ, ρ, λ, τ). The main advantage of using Rao s score test principal is that we need estimation of θ only under the joint null H a : σ μ = ρ = λ = τ = i.e., of θ = (β, σ e,,,,). We assume the weight matrices W and M to be same. This is often realistic in practice, since there may be good reasons to expect the structure of spatial dependence to be same for the dependent variable Y and the disturbance term ε. On the basis of the results described in the Appendix, Section, the score functions and the information matrix J evaluated under H a i.e., restricted ML estimator of θ, i.e. with γ = (β, σ e ) are : L β = (7) L σ e = (8) L σ = NT J T I N u μ σ u e u u (9) L ρ = NT (G I N )u u u () u L λ = NT I T W+W u u u () u L τ = NT σ e [u (I T W)u] () where u = y xβ is the OLS residual vector, and σ e = u u NT.

13 The information matrix J (Eq. ) under H a is X (I T I N )X σ e X (I T I N )Xβ σ e NT σ e 4 NT σ e 4 NT σ e 4 NT σ e 4 N(T ) σ e N(T ) σ e N(T ) Ttr(W + WW ) Ttr(W + WW ) X (I T I N )Xβ σ e Ttr(W + WW ) H where H = Ttr(W + WW ) + (I T I N )((M I T )Xβ) ((M I T )Xβ) σ, G is a bidiagonal matrix with e bidiagonal elements all equal to one, J T is T x T matrix of ones. Apart from the RS statistic for full joint null hypothesis H a, we also derive the modified test statistic for the following hypotheses: I) H b : σ μ = in presence of ρ, λ, τ II) H c : ρ = in presence of σ μ, λ, τ III) H d : λ = in presence of σ μ, ρ, τ IV) H e : τ = in presence of σ μ, ρ, λ. (3) 3

14 a As discussed earlier, this strategy will guide us to identify the correct source(s) of departure(s) from H when it is rejected. There is a big advantage in compiling all the test statistics only under the joint null. Given the full specification of the model in Eq. (7) - (), it is of course possible to derive conditional RS and likelihood ratio (LR) tests, for say, σ μ = in the presence of ρ, λ, τ as advocated in Baltagi et. al. (3), Baltagi et. al. (7) and Baltagi and Liu (8). However, that requires ML estimation of (ρ, λ, τ) (and also of σ μ for LR test), which could be difficult and cumbersome to obtain in some cases, especially in our general model framework. For example, as we note below for case I, i.e., for I) H b : σ μ = in presence of (ρ, λ, τ), the term J ψφ.γ i.e., J σμ φ.βσ e whereψ = σ μ and φ = (ρ, λ, τ). Thus the parameter σ μ is not independent of (ρ, λ, τ ) and vice- versa and therefore, the marginal RS test statistic, i.e. RS σμ for H b : σ μ = assuming φ = (ρ, λ, τ) = (,,) is not a valid test under the presence of, λ, ρ. Instead RS σμ which eliminates the effects of (τ, λ, ρ ) without estimating them, would be a more appropriate statistic, as discussed above. Therefore, the focus of our strategy is to carry out the specification test for a general model with minimum estimation. As we will see later from our Monte Carlo results, we lose very little in terms of finite sample size and power. Though RS σμ does not require explicit estimation of (τ, λ, ρ ), effect of these parameters have been taken into account through the use of the effective score d.φ σμ. We now discuss the test statistics for each of the above hypotheses. From Eq. (5) recall the form of locally size adjusted RS for H : ψ = in presence of parameter φ: RS ψ = d ψ θ J ψφ.γ θj φ.γ θ d φ θ J ψ.γ θ J ψφ.γ θj φ.γ θ J φψ.γ θ d ψ θ J ψφ.γ θj φ.γ θ d φ θ For each of the following test, γ = { β σ e } and the derivation of each test statistics are: (Detail derivations are in the Technical Appendix, Section II) 4

15 I) H b : σ μ = in presence of ρ, λ, τ. Here we are testing the significance of random location/individual effect in presence of time series autocorrelation of errors, spatial error dependence and spatial lag dependence. In particular we have ψ = σ μ and φ = [ ρ λ τ]. Here J ψφ.γ = [ J σμ ρ ] = [ N(T ) σ e ], which implies that the unadjusted RS is not a valid test under the local presence of [τ, λ, ρ]. It also means that only the partial covariance between d σμ and d ρ is nonzero, while it is zero for d σμ and d λ ; d σμ and d τ. The test statistic for H b has the following form: d σ μ J σ μ ρ J ρ d ρ RS σμ = (4) J σ μ.σe J σ μ ρ J ρ J ρσ μ In the numerator, [d σμ J σμ ρ J ρ d ρ ] is the part of d σμ that remains after eliminating the effect of d ρ, other nuisance parameters λ (spatial error lag) and τ(spatial dependence lag) have no (asymptotic effect) on the test for σ μ (individual/location random effect). Similar interpretation applies to the denominator, the variance part. Thus inference regarding σ μ is affected only by the presence of time-series autocorrelation ρ and is independent of the spatial aspects of the model. II) H c : ρ = in presence of σ μ, λ, τ. Here we are testing the significance of time-series autocorrelation in presence of random location/individual effect, spatial lag and spatial error dependence effects. For this test we have ψ = ρ and φ = [σ μ λ τ]. J ψφ.γ = [ J ρσμ ], where J ρσμ = N(T ), has two implications: σ e i) The unadjusted RS ρ is not valid under the local presence of the nuisance parameters. ii) Only the covariance between d ρ and d σμ is nonzero, while it is zero for d ρ and d λ ; and for d ρ and d τ. 5

16 Thus the new adjusted test statistic is RS ρ = d ρ J ρσ μ J σμ.σe d σ μ (5) [J ρ J ρσ μ J σμ.σe J σ μ ρ ] which adjusts the asymptotic mean and variance of the marginal RS statistic. The mean part reflects the effect of d ρ after eliminating the effect of the nuisance parameters. In this case, given the form of J ψφ.γ, the presence of d σμ among all other scores of nuisance parameters in the vector d φ, affects the test statistics. Thus inference on ρ is affected only by the presence of random effect, not by the presence of spatial dependence, given β and σ e. III) H d : λ = in presence of [σ μ, ρ, τ]. We are testing the significance of spatial error dependence in presence of random individual/location effects, error autocorrelation and spatial lag dependence effect. Thus, we have ψ = λ and φ = σ μ ρ τ. Here, J ψφ.γ = [ J λτ ], where J λτ = Ttr(W + WW ) which implies that partial covariance of d λ with d τ is nonzero; and it is zero for d λ with d σμ and d ρ respectively. It also indicates that the unadjusted RS λ is not valid under the local presence of nuisance parameters. The new adjusted test statistics is RS λ = d λ J λτ J τ.β d τ J λ J λτ J τ.β J τλ (6) which eliminates the effect of nuisance parameters. Now given the form of J ψφ.γ, only the presence of the spatial lag dependence τ affects the inference of spatial error dependence λ; and this is independent of the presence of so called time-series dimension of the model ie, random effects (σ μ ) and time-series autocorrelation effect(ρ). 6

17 IV) H e : τ = in presence of [σ μ, ρ, λ]. Here we are testing the significance of spatial lag dependence τ, given the local presence of random effect σ μ, autocorrelation ρ and spatial error dependence λ. Again, J ψφ.γ = [ J τλ ], where J τλ = Ttr(W + WW ) implies that the partial covariance of d τ and d λ is nonzero, while it is zero for d τ with d σμ and d ρ respectively. Thus it not only implies the invalidity of unadjusted RS τ, and leads to the adjusted RS test as RS τ = d τ J τλ J λ J τ.β J τλ J λ d λ J λτ. (7) As above, the inference of the spatial lag dependence is affected by the local presence of spatial error dependence; but not by other dimension of the model. The above tests indicates that inference regarding the time non-spatial dimensions of the model [σ μ and ρ] is not affected by the presence of the spatial counterpart (λ and τ ) and vice versa. This is consistent with the following partial covariances that are all zero. (Derivation of these partial covariance terms is provided in Technical Appendix, Section III): J ρ(λτ).σμ γ =, implying partial covariance of d ρ with d τ and d λ is zero after eliminating the effect of d σμ and d γ. J σμ (λτ).ργ =, implying partial covariance of d σμ with d τ and d λ is zero after eliminating the effect of d ρ and d γ. J λ(σμ ρ).τγ =, implying partial covariance of d λ with d ρ and d σμ is zero after eliminating the effect of d τ and d γ J τ(σμ ρ).λγ =, implying partial covariance of d τ with d ρ and d σμ is zero after eliminating the effect of d λ and d γ. Lastly, J (λτ)(σμ ρ).γ = implies that the partial covariance of (d σμ, d ρ ) with (d τ, d λ ) is zero after eliminating the effect of dγ. It is also evident from the joint four-dimensional RS test. By decomposing the joint RS for H a : σ μ = ρ = λ = τ =, we get the equivalent form as in Eq. (6): 7

18 RS σμ ρλτ = RS σμ ρ + RS λτ = RS σμ + RS ρ + RS τ + RS λ = RS σμ + RS ρ + RS τ + RS λ (8) The statistic is distributed as χ 4 () and will of course result in loss of power compared to the proper one-directional test as discussed above. Note that the statistic is not the sum of four marginal RS statistics because the interdependence within the spatial and non-spatial parameters. Infact, we can express it as RS σμ ρλτ = RS σμ ρ + RS λτ [(RS σμ + RS ρ ) d σμ R ( d ρ d σ μ ) d ρ ] + [(RS τ + RS λ ) d λ R (dλ d τ )d τ ] (9) where R ( d ρ d σ μ ) is the correlation of d ρ and d σμ ; and R (dλ d τ ) is the correlation of d λ and d τ. Thus the above result also support our finding that the unadjusted RS over rejects the null as it fails to take into account of the effect of the local presence of nuisance parameters. In particular they ignore the relevant interaction terms (Derivation is shown in Technical Appendix Section IV). It is only in case when the correlations between the scores of the parameters are independent of each other, then the unadjusted RS is asymptotically equivalent to adjusted RS. From the additive property above, one can easily obtain the adjusted RS tests from their unadjusted counterparts for each of the parameters without any extra derivations. For instance, we can get the RS σμ and RS ρ as RS σμ = RS σμ ρ RS ρ. (3) RS ρ = RS σμ ρ RS σμ (3) Similarly, RS τ = RS λτ RS λ (3) and RS λ = RS λτ RS τ. (33) This provides a great computational simplicity for the practitioners. One can easily obtain the joint RS(two directional) and marginal RS (one directional) for the parameters using any popular statistical programming package like STATA, R, Matlab just based on an OLS residuals. Using these, one can obtain the adjusted RS statistics as shown above. Thus our size adjusted robust methods are easily implementable without any kind of computational burden, unlike the conditional tests as derived in the literature. 8

19 Within our framework it is also easy to derive robust joint tests for example H f : σ μ = τ = in possible presence of ρ and λ. One testing strategy would be to apply RS σμ ρλτ first at say, α-level. If it is significant, then use RS σμ ρ and RS λτ separately at α/ level. Depending on which ones are significant, we can use the adjusted tests RS ρ, RS σμ, RS λ and RS τ at even lower level, say α/4. In future Monte Carlo study we will evaluate the performance of such sequential testing strategy. In the following section we report results for a simulation study using the marginal and adjusted tests in a straight forward manner. 5. Monte Carlo Results In this section we present the results of a Monte Carlo study to investigate the behavior of the finite sample behavior (size and power) of the tests. To facilitate comparisons with existing results we follow a structure close to Baltagi et al. (7) and Baltagi and Liu (8). The data were generated using eq. (7)-(): N y it = α + τ j= w ij y jt + X it β + u it, for i =,,, N; t =,,, T, (7) u it = μ i + ε it (8) ε it = λ N j= w ij ε jt + v it (9) v it = ρv it + e it, where e it ~IIDN(, σ e ). () We set α = 5 and β =.5. The independent variable X it was generated following Nerlove(97): X it =.t +.5X it + ω it, where ω it has the uniform distribution on [-.5,.5]. Initial values were chosen as in Baltagi et al(9). We consider both the rook and the queen design for the weight matrix W. We fix σ μ + σ e = and let η = σ μ /(σ μ + σe ) vary over a range from to.5 with an increment of.5. The spatial lag and error dependence parameters i.e., τ, λ respectively and also ρ are varied over a range from to.5 with an increment of.5. Three different pairs (N,T) are considered for each rook design W matrix and queen W matrix : 9

20 {(5,),(49,),(49,)}. For each sample size, we generate samples for each different parameter settings. Therefore maximum standard errors of the estimates of the size and power would be.5(.5)/.5. In each replication the parameters were estimated using OLS, and eleven test statistics, namely, RS σμ ρλτ, RS σμ ρ, RS λτ, RS σμ, RS ρ, RS λ, RS τ, RS σμ, RS ρ, RS τ and RS λ were computed. The tables and graphs are based on the nominal size of.5. To save space we present only a portion of our results. We have repeated the experiments for RS ρ and RS σμ for the following cases i) λ = and τ = (When the spatial dimensions are absent; this case is similar as in Bera et.al ()). ii) λ =.5 and τ =.3 (i.e., when there is local presence of the spatial parameters. Now assuming λ = and τ, would make our experiment results comparable to Rojas(), Baltagi and Liu (8) ). iii) λ =.3 and τ =.5(Assuming λ and τ = would make our experiment results comparable to Baltagi et. al (7).) Similarly for RS τ and RS λ we have repeated our experiments with i) ρ = and σ μ = (When the time series dimensions are absent; this case is similar to Anselin et al(996)). ii) ρ =.5 and σ μ =.3 iii) ρ =.3 and σ μ =.5 Let us now turn into the performances of the tests in terms of power and sizes of RS ρ and RS σμ. For N=5, T=, the estimated rejection probabilities are reported in Table, and for N=49, T= it is reported in Table. For both of these tables, the estimated rejection probabilities are for data generated with λ = and τ =. We also illustrate the results of the other two cases graphically from Figures -8. Moreover, our adjusted tests are designed for locally misspecified alternatives close to ρ =, σ μ = and λ =, τ =. The main objective of our Monte Carlo study is to investigate the performance of our suggested tests in the neighborhood of these parameter values. Let us first concentrate on RS σμ and RS σμ, which are

21 designed to test H : σ μ =. When ρ =, there is a loss of power for RS σμ vis-a-vis RS σμ, and this loss gets minimized as η deviates more and more from zero. While RS σμ doesnot sustain much loss in power when ρ =, we notice that RS σμ reject H : σ μ = too often when σ μ =, but ρ. This unwanted rejection probabilities is due to the noncentrality parameter, as discussed by Bera et. al. (). RS σμ also has some rejection probabilities but the problem is less severe. Moreover, even when we allow for case ii) =.5 and τ =.3; case iii) λ =.3 and τ =.5, we find similar results, i.e., RS σμ are robust only under the local misspecification, i.e., for low values of ρ. From Figures ) and ) we can see that RS σμ is size robust for local misspecification of the parameters under both the cases ii) λ =.5 and τ =.3; iii) λ =.3 and τ =.5. Comparing the power (Figures 3-4), it is clearly evident that the power loss gets minimized for RS σμ as η deviates from. Looking at the comparison of the rejection probabilities, i.e., Figure (5) - (6) for λ =.5 and τ =.3 ; and Figures (7) (8) for λ =.3 and τ =.5, it is evident that when η >, increase in ρ enhances the rejection probabilities of RS σμ. This is due to the fact that the noncentrality parameter of RS σμ depends on ρ, whereas for RS σμ, it doesnot depend on ρ. This result is consistent with Bera et. al. ().

22 Table Estimated rejection probabilities, with λ =, τ =. Sample Size: N=5,T= η ρ RS σμ ρλτ RS σ μ ρ RS ρ RS ρ RS σμ. RS σμ

23 Table Estimated rejection probabilities, with λ =, τ =. Sample Size: N=49,T= η ρ RS σμ ρλτ RS σ μ ρ RS ρ RS ρ RS σμ. RS σμ

24 Size Comparison of RS(mu) and RS(mu*) ) λ =.5 and τ =.3 ) λ =.3 and τ =.5 Estimated Size Rho RS(mu*) RS(mu) Estimated Size Rho RS(mu*) RS(mu) Power Comparison 3) λ =.5 and τ =.3 4) λ =.3 and τ =.5 Estimated Power RS(mu*) RS(mu) Estimated Power RS(mu*) RS(mu) sigmamu sigmamu 5) Rejection Probabilities of RS(mu*) 6) Rejection Probabilities of RS(mu) λ =.5 and τ =.3 Rejection Probabbilities sigmamu rho= rho=. 5 rho=. rho=. Rejection Probabilities rho= rho=.5 rho=. rho=. rho= sigmamu 4

25 λ =.3 and τ =.5 7 ) Rejection Probabilities of RS(mu*) 8) Rejection Probabilities of RS(mu). Rejection Probabilities rho= rho=.5 rho=. rho=. rho= sigmamu Rejection Probabilities sigmamu rho= rho=.5 rho=. rho=. rho=.3 As shown mathematically, the figures -8 confirm that the local presence of the spatial dimensions do not affect RS σμ drastically. The features of RS σμ is more or less similar when λ = and τ = verses when their local departures from. It means that the inference of the parameter σ μ does not depend on the local presence of λ and τ. In similar way, we can explain the behavior of RS ρ using Table and and figures 9-6. From Table and we note that when σ μ = then RS ρ has better power than RS ρ. However, unlike RS σμ, the power of RS ρ is much closer to RS ρ. The real benefit of RS ρ is noticed when ρ = but η > ; the performance of RS ρ is remarkable. For the case λ =, τ =, for N=5, T= and N=49, T= it is evident from the Tables and. Even when there is local presence of the parameters λ and τ, the size of RS ρ is significantly better than RS ρ when η >. In other words, RS ρ is performing much better than what it is expected to perform; i.e. not rejecting ρ = when ρ is indeed zero even for large values of η. On the other hand RS ρ rejects the null too often even when ρ is actually zero. The power comparison also gives a very impressive result. We find that RS ρ is strongly affected by the presence of the random effect, while there is very little effect on the power of RS ρ as seen from figures

26 Size Comparison of RS(rho) and RS(rho*) 9) λ =.5 and τ =.3 ) λ =.3 and τ = Estimated Size.6.4. RS(rho*) RS(rho) Estimated Size.6.4. RS(rho*) RS(rho) sigmamu sigmamu Power Comparison ) λ =.5 and τ =.3 ) λ =.3 and τ =.5 Estimated Power RS(rho*) RS(rho) Estimated Power RS(rho*) RS(rho) rho rho 3) Rejection Probabilities of RS(rho*) 4) Rejection Probabilities of RS(rho) λ =.5 and τ =.3 Rejection Probabilities sigmamu= sigmamu=.5 sigmamu=. sigmamu= rho Rejection Probabilities sigmamu= sigmamu=.5 sigmamu=. sigmamu= rho 6

27 λ =.3 and τ =.5 5) Rejection Probabilities of RS(rho*) 6) Rejection Probabilities of RS(rho) Rejection Probabilities sigmamu= sigmamu=.5 sigmamu=. sigmamu= rho Rjection Probabilities sigmamu= sigmamu=.5 sigmamu=. sigmamu= rho The performance of the joint statistics RS σμ ρλτ and RS σμ ρ is optimal under the optimal condition, i.e., when σ μ = ρ = λ = τ = and when σ μ = ρ = respectively. However, if there is departure in one, two or three directional way, then RS σμ ρλτ and RS σμ ρ fail to identify the exact source of misspecification. However, overall they have good power. These results are consistent with Bera et. al. () and also Rojas (). However, we differ from each of them in our basic model framework, which is more general than both Bera et. al. () and Rojas (). Our size-adjusted robust test statistic is much easier to compute then the one dimensional or two dimensional conditional tests as proposed by Baltagi et. al. (7). From the performance perspective also, our test statistics do equally good than the conditional LM tests as proposed by later. Let us now consider the parameters of spatial dimensions. RS τ and RS λ are affected by λ and τ respectively, and not by the presence of η which is a function of σ μ and ρ. To explore the performance of these tests we have performed the Monte Carlo study for three cases: i) η =ρ= (This case is exactly similar to Anselin et. al.(996), and our results are comparable to their findings.) ii) η=.5 and ρ=.3 iii) η=.3 and ρ=.5 7

28 Table 3 Estimated Rejection probabilities with η = ρ =, Sample (N,T)=(5,) τ λ RS σμ ρλτ RS λτ RS λ RS λ RS τ RS τ

29 Table4: Estimated Rejection probabilities with η = ρ =, Sample (N,T)=(49,) τ λ RS σμ ρλτ RS λτ RS λ RS λ RS τ RS τ

30 Table 3 and 4 gives the estimated rejection probabilities of the tests RS λ, RS λ, RS τ, RSτ RS σμ ρλτ, and RS λτ, for the case η = ρ = for sample size (5,) and(49,) respectively. The RS λ has power against a spatial lag, although less than the lag tests i.e., RS τ. The behavior of RS λ is interesting. It has no power against lag dependence i.e. τ, as it should. For small values of τ, the rejection frequency of RS λ is very close to its expected value of.5. Infact for (N, T =49,) this size robustness of RS λ is more evident as the rejection frequency is close to.5 even when τ =.5. In other words, RS λ does its job very well, even more than what it is designed to do for. However the rejection frequency of RS λ is large in presence of τ even when λ is actually equal to zero. This reiterates our result that RS λ is robust to local misspecification, while the test results of RS λ can be very misleading in presence of such nuisance parameters. In terms of power, RS λ is trailing just behind RS λ as can be clearly seen from the tables. This slight loss in power is due to the presence of the non centrality parameter, as discussed by Bera and Yoon (993). We further investigated the behavior of RS λ for the two other cases also: ii) η=.5 and ρ=.3 iii) η=.3 and ρ=.5 The results are explained through the figures 7-4. The size of RS λ is much better than its unadjusted counterpart in local presence of all the three parameters τ, η and ρ. The power of RS λ is slightly less than that of RS λ, given τ = for both the above cases. The figures -4, clearly show that the rejection probabilities are very close to.5 for RS λ for τ varying from,.5,.,.3 and.5. The rejection probability of RS λ increases with λ as it should be. On the other hand the rejection probability of RS λ is always very high even when λ = and τ being away from zero. This re-iterates our earlier result as evident from Table 3 and 4, when both the random region effect and error autocorrelation effect were absent ie, η= ρ=.. These experimental results provide further support to our mathematical findings. 3

31 Size Comparison of RS(lamda) and RS(lamda*) 7) η =.5 and ρ =.3 8) η =.3 and ρ =.5 Estimated Size RS(lamda*) RS(lamda) Estimated Size RS(lamda*) RS(lamda) tau tau Power Comparison 9) η =.5 and ρ =.3 ) η =.3 and ρ =.5 Estimated Power lamda RS(lamda*) RS(lamda) Estimated Power RS(lamda*) RS(lamda) lamda ) Rejection Probabilities of RS(lamda*) ) Rejection Probabilities of RS(lamda) η =.5 and ρ =.3 Rejection Probability tau= tau=.5 tau=. tau=.3 tau=.5 Rejection Probability tau= tau=.5 tau=. tau=.3 tau=.5 lamda lamda 3

32 η =.3 and ρ =.5 3) Rejection Probabilities of RS(lamda*) 4) Rejection Probabilities of RS(lamda) Rejection Probability tau= tau=.5 tau=. tau=.3 tau=.5 Rejection Probability tau= tau=.5 tau=. tau=.3 tau= lamda lamda Finally, we discuss the experimental results of RS τ and RS τ. When η = ρ =. as in Tables 3 and 4, RS τ is size robust for local misspecification of λ. For λ > and τ =, RS τ has rejection probabilities higher than.5, but it is much less than RS τ. This unwanted rejection probabilities of RS τ is due to the noncentrality term which depends on λ. As mentioned before, RS τ is designed to be robust only under local misspecification, i.e., for low values of λ. From that point of view, it does a good job; the performances detoriate as λ takes higher values. From the Tables we also note that when τ > an increase in λ enhances the rejection probabilities of RS τ. This is due to the presence of λ in the noncentrality parameter of RS τ. But the noncentrality parameter of RS τ does not depend on λ (Proofs regarding non centrality parameters of these tests can be found in Bera and Yoon (993) ).This result is valid only asymptotically and for local departures of λ. We further investigate the behavior of these tests for the cases when ii) η=.5 and ρ=.3 iii) η=.3 and ρ=.5 Figures 5-3 explains the results of the tests for the above two cases. 3

33 The size of the test RS τ is robust for the local presence of λ; and it increases to approximately.4 when λ =.5. In contrast the size of RS τ is approaches when λ =.5. Thus although there is some unwanted rejection probability problem with RS τ (as discussed before) still the problem is much less severe of RS τ than RS τ ; even when there is local presence of η and ρ. The power of RS τ trails behind RS τ, but becomes c lose to each other for larger values of τ. Comparing the rejection probabilities of RS τ verses RS τ (Figures 9-3) we find that presence of λ enhances the rejection probability of the former more than the latter. Since λ does not influence the non centrality parameter of RS τ, so the rejection probabilities of RS τ for λ =,.5,.,.3 is close to each other, and they are close to when τ =.5. However for RS τ, as λ increases the rejection probabilities increases and is infact close to when λ =.3 and τ =. These results is in similar lines of what we found when η and ρ were zero, i.e., the local presence of the random region effect and time series error autocorrelation do not influence the inference of these tests. Thus these once again support our mathematical finding regarding these tests. Size Comparison of RS(tau) and RS(tau*) 5) η =.5 and ρ =.3 6) η =.3 and ρ =.5 Estimated Size RS(tau*) RS(tau) lamda Estimated Size RS(tau*) RS(tau) lamda 33

34 Power Comparison RS(tau) and RS(tau*) 7) η =.5 and ρ =.3 8) η =.3 and ρ =.5 Estimated Power RS(tau*) RS(tau) tau Estimated Power RS(tau*) RS(tau) tau 9) Rejection Probabilities of RS(tau*) 3) Rejection Probabilities of RS(tau) η =.5 and ρ =.3 Rejection Probability lamda= lamda=.5 lamda=. lamda-. lamda= tau Rejection Probability lamda= lamda=.5 lamda=. lamda-. lamda= tau η =.3 and ρ =.5 3) Rejection Probabilities of RS(tau*) 3) Rejection Probabilities of RS(tau) Rejection Probability lamda= lamda=.5 lamda=. lamda-. lamda= tau Rejection Probability lamda= lamda=.5 lamda=. lamda-. lamda= tau 34

35 One important thing to note is these one dimensional size robust are more meaningful not only from their marginal counterparts but also from the joint tests, RS σμ ρλτ (four dimensional), RS λτ and RS σμ ρ(two dimensional) tests. These joint tests are only optimal when σ μ = ρ = λ = τ =. These tests fail to identify the exact source of misspecifications. This is evident from Tables -4. The results of RS τ and RS λ for the case i) η= ρ =. is consistent with Anselin, Bera et al (996); however we differ from them as our model is more extensive and complete than what the authors considered. Moreover, we have also explored these tests when there is misspecification in more than one nuisance parameters. In addition, as stated before our conditional tests not only give intuitive results which one can explain analytically and also mathematically, but these are also easy to compute (they are all based on OLS estimates) than the tests proposed by Baltagi et al (7). Infact one can easily derive the adjusted test statistic using the simple unadjusted ones as shown in our paper in the previous section. Thus our tests are much more amenable to model practioners. 6. Empirical application to growth theory models One of the traditional stylized facts about growth over the last 5 years is that national growth rates appear to depend critically on the growth rates and income levels of other countries, rather than just on any one country s own domestic investment rates in physical and human capital. For example, Easterly and Levine () present as a stylized fact the concentration of economic activity at different scales: world, countries, regions, cities. More recently, Klenow and Rodriguez-Clare (5) present stylized facts reflecting worldwide interdependence, which could be explained by cross-country externalities. Several models of economic growth emphasize the importance of international spillovers as a major engine of technological progress. These international spillovers result from foreign knowledge through international trade and foreign direct investment (Coe and Helpman, 995; Caves, 996), or technology transfers (Barro and Sala-i-Martin, 997; Howitt, ) or human capital externalities (Lucas 988, 993). Temple (999), in his survey of the new growth evidence, draws attention to error correlation and regional spillovers, though he interprets these effects as mainly reflecting an omitted variable. The empirical results of this section are yet to be completed. 35