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A simple spatial dependence test robust to local and distributional misspecifications
Economics Letters 124 (2014) 203–206
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A simple spatial dependence test robust to local and distributional misspecifications✩ Ying Fang a , Sung Y. Park b,∗ , Jinfeng Zhang c a
The Wang Yanan Institute for Studies in Economics, MOE Key Laboratory of Econometrics, Fujian Key Laboratory of Statistic Science, Xiamen University, Xiamen, Fujian 361005, China b
School of Economics, Chung-Ang University, 84 Heuksok-Ro, Dongjak-Gu, Seoul, Republic of Korea
c
The Center for Economic Research, Shandong University, Jinan, Shandong 250100, China
highlights • We propose a test for spatial dependence which is robust to local misspecification and distributional misspecification. • We find that Burridge (1980)’s test is robust to distributional misspecification. • We find that Anselin, Bera, Florax and Yoon (1996)’s test is robust to distributional misspecification.
article
info
Article history: Received 26 January 2014 Received in revised form 19 May 2014 Accepted 20 May 2014 Available online 28 May 2014
abstract In this paper, we derive a class of modified score tests robust to local and distributional misspecifications for testing spatial error autocorrelation and spatial lag dependence. The proposed tests are general enough to include several popular tests for the spatial dependence as special cases. Moreover, we show that the popular test statistics proposed by Burridge (1980) and Anselin et al. (1996) are robust to distributional misspecification although they are derived under normality assumption. © 2014 Elsevier B.V. All rights reserved.
JEL classification: C12 C21 R10 Keywords: Spatial dependence Score test Robust test Distribution misspecification Local misspecification
1. Introduction Moran (1950) proposes a seminal test, Moran’s I-test, for spatial autocorrelation in a regression model framework, but the test does not provide the nature of the spatial process that causes spatial autocorrelation, particularly, whether the spatial dependence is due to an autoregressive error process or omitted spatially lagged dependent variables. Burridge (1980) extends Moran’s I-test based on
✩ We thank the editor Badi Baltagi and one anonymous referee for their valuable comments. Fang acknowledges financial supports from NSFC 71271179, NSFC 71131008 and NCET-12-0328. Zhang acknowledges financial supports from NSFC 71301088. All remaining errors are ours. ∗ Corresponding author. Tel.: +82 2 820 5622; fax: +82 2 812 9718. E-mail address: [email protected] (S.Y. Park).
http://dx.doi.org/10.1016/j.econlet.2014.05.015 0165-1765/© 2014 Elsevier B.V. All rights reserved.
the Lagrange multiplier (LM) principle to test the spatial error autocorrelation without considering the presence of spatially lagged dependent variables, and furthermore, Anselin (1988b) proposes a LM test for the spatial error autocorrelation in the case of the presence of the spatially lagged dependent variable. However, the test involves a nonlinear optimization or applies a numerical search technique. Anselin et al. (1996) propose a modified score test for the spatial error autocorrelation in the presence of local misspecification to the parameter corresponding to the spatial lag dependence. Comparing to the Ansenlin’s LM test, the latter only requires the ordinary least squares (OLS) residuals under the null hypothesis and has little computational burden (Bera and Bilias, 2001). Yet one potential problem of these tests is that the underlying probability density may not be correctly specified, i.e., there may exist the distributional misspecification problem.
204
Y. Fang et al. / Economics Letters 124 (2014) 203–206
We develop a modified score test for spatial dependence, robust to both the local and distributional misspecifications. Local parametric misspecification arises when some nuisance parameters deviate locally from their true values. Distributional misspecification occurs when the underlying data generating process (DGP) is not correctly specified. When nuisance parameters are locally deviated from their true values, the score statistic has a non-zero drift term in general (Davidson and MacKinnon, 1987; Saikkonen, 1989). Bera and Yoon (1993) propose a modified score test robust to the local misspecification. On the other hand, when the underlying probability distribution is misspecified, some standard results are not valid any more. For example, the information matrix (IM) equality is invalid under distributional misspecification. Making inferences with the distributional misspecification can cause size distortion of the test statistics. For example, White (1982) suggests a modified LM test based on the restricted quasi-maximum likelihood (QML) estimator to deal with the distributional misspecification. Bera et al. (2007) propose a score test that is not only robust to local misspecification but also to distributional misspecification in the spirit of White (1982) and Bera and Yoon (1993). Recently, Baltagi and Yang (2013a,b) modify the standard LM test for spatial error dependence robust against distributional misspecification for cross-sectional and panel data models, respectively. In the spatial econometrics literature, the corresponding robust score (LM) test which deals with the distributional misspecification or both the local and distributional misspecifications have not been studied yet. There are some studies that suggest some other types of robust tests. However, their test statistics are based on other different estimation methods, and neither of them considers the local misspecification.1 A modified test can be constructed by properly adjusting the mean and variance of the usual score test statistics, and therefore, it has correct asymptotic size. The test statistics can be simplified when either the error term follows the normal distribution or the nuisance parameters are estimated consistently. In the next section, we first review spatial dependence tests in a spatial autoregressive model with a spatial autoregressive disturbance. We then develop new score tests robust to both local parametric and distributional misspecifications. The paper concludes in Section 3.
following Anselin (1988a), the log-likelihood function of (2.1) and (2.2) is given by N
ln 2π −
(2.1) (2.2)
where y is the dependent variable, X is a N × k matrix of explanatory variables, β denotes a k × 1 unknown parameter vector, ρ and λ are scalar spatial parameters, W1 and W2 are N × N known spatial weight matrices, ϵ is an N × 1 vector of regression disturbances, and u is an N × 1 vector of innovations with ui ∼ i.i.d(0, σ 2 ) for i = 1, 2, . . . , N. For the notational convenience, we denote θ = (β ′ , σ 2 , λ, ρ)′ , γ = (β ′ , σ 2 , λ)′ , η = (β ′ , σ 2 )′ , A = IN − ρ W1 , B = IN − λW2 , GA = W1 A−1 and GB = W2 B−1 . Under the normality assumption,
RSλ =
u˜ ′ W2 u˜ /σ˜ 2
2 ,
T22
(2.4)
where T22 = tr (W2 + W2′ )W2 , u˜ = y − X β˜ and σ˜ 2 = u˜ ′ u˜ /N.
Here θ˜ = (β˜ ′ , σ˜ 2 , 0, 0)′ denotes the constraint maximum likelihood estimator (MLE) under H0λ . The above statistic RSλ converges in distribution to χ12 under H0λ . However, if the nuisance parameter ρ ̸= 0, for example,√which is contaminated by a local deviation such that ρ = δ1 / N where δ1 is a nonzero finite constant, the Burridge statistic converges to a noncentral chi-square distribution, which implies that RSλ generally over-reject the null hypothesis even if H0λ is true. Anselin et al. (1996) construct a robust score test for the spatial error autocorrelation by eliminating the noncentral term. Using the one-step method-of-scoring estimator, the modified score test is given by
u˜ ′ W2 u˜ /σ˜ 2 − T21 N J˜ρ·η
−1
2 T22 − T21 N J˜ρ·η
u˜ ′ W1 y/σ˜ 2
2 ,
−1
(2.5)
where u˜ are the OLS residuals, σ˜ 2 = u˜ ′ u˜ /N, T21 = tr (W2 +
W2′ )W1 , J˜ρ·η =
1 N σ˜ 2
T11 σ˜ 2 +
W1 X β˜
′
MX (W1 X β) for T11 =
−1
tr (W1 + W1′ )W1 and MX = IN −X X ′ X X . RSλL converges to χ12 λ distribution under H0 even though ρ deviates locally from 0 such
√
that ρ = δ1 / N. We consider a modified score test robust to both local and distributional misspecifications, i.e., g (y) ̸= f (y, θ ) and ρ = ρ0 + √ δ1 / N where ρ0 is a finite constant and δ1 > 0. Under H0λ : λ = 0, the modified score test is given by RSλLD
−1 2 −tr(GA ) + u˜ ′ W1 y/σ˜ 2
−1
u˜ ′ W2 u˜ /σ˜ 2 − T˜2A N J˜ρ·η
= 2 N J˜ρ·η T22 − T˜2A
1 An incomplete list includes, Anselin (1990), Anselin and Kelejian (1997), Kelejian and Robinson (1998), Anselin and Moreno (2003), Saavedra (2003) and Yang (2010) among others.
1
For testing the spatial error dependence, the null hypothesis of interest is H0λ : λ = 0. Burridge (1980) proposes a one-directional score test for H0λ by assuming ρ = 0 in (2.1):
RSλL =
y = ρ W1 y + X β + ϵ, ϵ = λW2 ϵ + u,
ln σ 2 + ln |A| + ln |B| −
2.1. A modified score test for spatial error dependence
2. A modified score test robust to local and distributional misspecifications Consider the following spatial autoregressive model with a spatial autoregressive disturbance (SARAR) model (Anselin, 1988a; Case, 1991; Anselin et al., 1996; Kelejian and Prucha, 2010; among others):
N
u′ u, (2.3) 2 2 2σ 2 where u = B(Ay − X β). The test statistics previously proposed by Burridge (1980), Anselin (1988b), and Anselin et al. (1996) are derived under the normality assumption of the error terms. However, when the distribution is misspecified, all tests are generally invalid and may have the size distortion problem. Let g (y) and f (y, θ ) be the true model and the specified model, respectively. When g (y) ̸= f (y, θ ), the information matrix K (θ ) and the negative expected Hessian matrix J (θ ) are not equivalent any more; see Appendix A for details. Hence, the variance–covariance matrix of the score statistic should be modified. In general, not only the mean but also the variance of the score test statistic have to be adjusted accordingly to take care of the local and distributional misspecifications. ln L(θ ) = −
2 −2 B ˜ ∗ρ·η + N1 T˜2A J˜ρ·η
, (2.6)
where the OLS residual u˜ = y − ρ0 W1 y − X β˜ , σ˜ 2 = u˜ ′ u˜ /N and ˜ ′ T˜2A = tr (W2′ + W2 )GA , J˜ρ·η = N1 [T˜AA − N2 tr 2 (GA ) + σ12 (GA X β)
Y. Fang et al. / Economics Letters 124 (2014) 203–206
˜ , B˜ ∗ρ·η = 1 4 [2µ ˜ ′ M G¯ + κ˜ G¯ ′ G¯ ]. Here A = MX (GA X β)] ˜ 3 (GA X β) Nσ ′ X A 4 A A −1 ˜ ¯ A = v ecD (GA )− IN −ρ0 W1 , GA = W1 A , TAA = tr GA + GA GA , G ( ) µ ˜ 3 = N1 Ni=1 u˜ 3i , µ ˜ 4 = N1 Ni=1 u˜ 4i and κ˜ 4 = µ ˜ 4 − 3σ˜ 4 , a N × 1 vector of ones and v ecD (GA ) denotes a column
1 l tr GA , N N lN denotes
vector formed by the diagonal elements of GA ; see Appendix B for details.2 The modified score test statistic converges to χ12 distribution under the null hypothesis. √ Now consider a special case ρ0 = 0, i.e., ρ = δ1 / n. This yields T˜2A = T21 , tr(GA ) = 0 and B∗ρ·η = 0, which implies that Anselin et al. (1996)’s test statistics given in (2.5) is automatically robust to distributional misspecification. Moreover, when ρ = 0 in the absence of the local parametric deviation, we have Jλη = 01×(k+1) , and then Kλ = Jλ and Bλ·η = Jλ·η under H0λ . Hence, the RSλLD in (2.6) is reduced to Burridge (1980)’s test statistic in (2.4). In other words, the Burridge test statistic (RSλ ) is also robust to the distributional misspecification when ρ = 0 and λ = 0. The reason is that, even though g (y) is different from the normal distribution, ρ0 = 0 yields B∗ρ·η = 0 and tr(GA ) = 0. Therefore, the distributional misspecification does not alter the form of test statistics of Burridge (1980) and Anselin et al. (1996). Additionally, we consider the case that ρ is an unknown parameter to be estimated under the null hypothesis H0λ . Since
(β˜ ′ , σ˜ 2 , ρ) ˜ ′ is the constrained MLE under H0λ , the score term (−tr(GA ) + u˜ ′ W1 y/σ˜ 2 ) is zero. Therefore, RSλLD can be given by 2 ′ u˜ W2 u˜ /σ˜ 2 LD RS λ = (2.7) −1 . 2 2 −2 B ˜ ∗ρ·η T22 − T˜2A N J˜ρ·η + N1 T˜2A J˜ρ·η Note that Var (ρ) ˜ = N J˜ρ·η in this case. Compared to Anselin
(1988b), the only difference comes from the fact that we correct the variance of the score test statistic in (2.7) to take care of the distributional misspecification. We conduct Monte Carlo simulations to examine the finite sample performance of the proposed tests.3 For the data generating process, we consider four different distributions of u in (2.2), normal, Student’s t, gamma and asymmetric bimodal mixture normal distributions, and three choices of weight matrices including the queen contiguity weight, the circular weight and the Anselin weight. We compare the size performances of RSλ and RSλLD under different distributions and different values of ρ . Note that in this case our proposed test RSλLD is identical to RSλL . The empirical sizes of both tests are quite correct when there exists only distributional misspecification. This is due to the robustness of both statistics. However, the actual sizes of RSλ increase quickly in the presence of local misspecification, while the size performance of RSλLD is relatively stable. We also compare
205
2.2. A modified score test for spatial autoregressive dependence A modified score test robust to local and distributional misspecifications for testing spatial autoregressive dependence can be derived as in (2.6), and the detail is given in Appendix C.4 Under ρ the null hypothesis H0 : ρ = 0, the modified score test is written by RSρLD
=
u˜ ′ BW1 y/σ˜ 2 − T˜BC1 (N J˜ρ·η )−1 −tr(GB ) + u˜ ′ GB u˜ /σ˜ 2
N J˜ρ·η −
˜
1 2 T N BC1
2
(J˜λ·η )−1 + N B˜ ∗ρ·η + C˜ ρ∗
, (2.8)
˜ and σ˜ 2 = u˜ ′ u˜ /N. T˜BC1 = where u˜ = (I − λ0 W2 )(y − X β) tr (G′B + GB )C1 , J˜λ·η = N12 NTBB − 2tr 2 (GB ) , J˜ρ·η = N1 [TC1 C1 + 1 ˜ ′ MBX (BW1 X β)] ˜ , TC1 C1 = tr (C1′ + C1 )C1 , C1 = (BW1 X β) σ˜ 2
BW1 B−1 , B˜ ∗ρ·η + C˜ ρ∗ = B = IN − λ0 W2 , GB J˜ρλ·η =
1 T N BC1
′ ˜ ′ MBX F˜ . Here κ˜ 4 F˜ F˜ + 2µ ˜ 3 (BW1 X β) −1 ˜¯ = W2 B−1 , F˜ = v ec D (C1 ) − J˜ρλ·η J˜λ·η GB , 1 N σ˜ 4
˜¯ = v ec (G ) − and G B D B
(GB ).
1 l tr N N
Similar to the case of RSλLD , i.e., λ = 0 in (2.2) the RSρLD in (2.8) is reduced to Anselin (1988b)’s equation (32) when λ is absent in the model. When λ0 = 0, the drift term in (2.8) is the same as that in Anselin et al. (1996). Moreover, since F˜ = 0 yields B˜ ∗ρ·η + C˜ ρ∗ = 0 in this case, the denominator in (2.8) is also the same as the denominator of equation 14 of Anselin et al. (1996), which implies equation 14 of Anselin et al. (1996), is automatically robust to the distributional misspecification when λ0 = 0. We also consider the case that λ is a parameter to be estimated ρ ˜ ′ is the constrained MLE under the null under H0 . Since (β˜ ′ , σ˜ 2 , λ) ρ
H0 , the score term S˜λ1 = 0. Therefore, the RSρLD is given by u˜ ′ BW1 y/σ˜ 2
LD
RS ρ =
N J˜ρ·η −
˜
1 2 T N BC1
2
. (J˜λ·η )−1 + N B˜ ∗ρ·η + C˜ ρ∗
(2.9)
Compared to Anselin (1988b), we adjust only the variance of the score test in (2.9) to correct the distributional misspecification. We also perform the similar Monte-Carlo simulations to anaLD
lyze the empirical sizes and powers of RSρ , RSρL , RSρLD and RS ρ using
the same simulation designs for the spatial error dependence case.5 The simulation results are quite similar with those of the spatial error dependence case. 3. Conclusion
LD
Anselin (1988b)’s test to our proposed test RS λ in the case that ρ is unknown and ρ0 ̸= 0. Since the nuisance parameter for both tests needs to be estimated, the actual sizes of both tests are robust to local misspecification. However, the impact of distributional misspecification on empirical sizes seems to be small. Finally, the power performance of all tests looks very good for all experiment designs. Our proposed test RSλLD achieves considerable powers when the model is either locally or globally misspecified. Moreover, RSλLD is more powerful than Anselin (1988b)’s test when there exist local and distributional misspecifications.
2 The Appendix can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf. 3 To conserve space we do not report the Monte-Carlo simulation designs and results but these can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf.
In this paper we derive modified score tests robust to both local and distributional misspecifications in the spatial autoregressive model with a spatial autoregressive error term. We show that some popular spatial dependence tests, such as Burridge (1980) and Anselin et al. (1996), can be expressed as special cases of our tests. We also find that Burridge (1980) and Anselin et al. (1996)’s tests are automatically robust to distributional misspecification under some special cases. Our findings in this paper support the usage of Anselin et al. (1996)’s tests for spatial dependence since they are also robust to distributional misspecification.
4 The Appendix can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf. 5 The test statistics, RS and RS L , are similar to RS and RS L , respectively, and can ρ
ρ
λ
be found in Burridge (1980) and Anselin et al. (1996).
λ
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Y. Fang et al. / Economics Letters 124 (2014) 203–206
Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2014.05.015. References Anselin, L., 1988a. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., 1988b. Lagrange multiplier test diagnostics for spatial dependence and spatial heterogeneity. Geogr. Anal. 20, 1–17. Anselin, L., 1990. Some robust approaches to testing and estimation in spatial econometrics. Reg. Sci. Urban Econ. 20, 141–163. Anselin, L., Bera, A.K., Florax, R., Yoon, M.J., 1996. Simple diagnostic tests for spatial dependence. Reg. Sci. Urban Econ. 26, 77–104. Anselin, L., Kelejian, H.H., 1997. Testing for spatial error autocorrelation in the presence of endogenous regressors. Int. Reg. Sci. Rev. 20, 153–182. Anselin, L., Moreno, R., 2003. Properties of tests for spatial error components. Reg. Sci. Urban Econ. 33, 595–618. Baltagi, B.H., Yang, Z., 2013a. Heteroskedasticity and non-normality robust LM tests for spatial dependence. Reg. Sci. Urban Econ. 43, 725–739. Baltagi, B.H., Yang, Z., 2013b. Standardized LM tests for spatial error dependence in linear or panel regressions. Econom. J. 16, 103–134. Bera, A.K., Bilias, Y., 2001. Rao’s score, Neyman’s C(α ) and Silvey’s LM tests: an essay on historical developments and new results. J. Statist. Plann. Inference 97, 9–44.
Bera, A.K., Bilias, Y., Yoon, M.J., Adjustments of Rao’s score test for distributional and local parametric misspecifications, 2007. Working paper, University of Illinoise at Urbana-Champaign. Bera, A.K., Yoon, M.J., 1993. Specification testing with locally misspecified alternatives. Econometric Theory 9, 649–658. Burridge, P., 1980. On the Cliff-Ord test for spatial correlation. J. Roy. Statist. Soc. Ser. B 42, 107–108. Case, A.C., 1991. Spatial patterns in household demand. Econometrica 59, 953–966. Davidson, R., MacKinnon, J.G., 1987. Implicit alternatives and the local power of test statistics. Econometrica 55, 1305–1329. Kelejian, H.H., Prucha, I.R., 2010. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. J. Econometrics 157, 53–67. Kelejian, H.H., Robinson, D., 1998. A suggested test for spatial autocorrelation and/or heteroskedasticity and corresponding Monte Carlo results. Reg. Sci. Urban Econ. 28, 389–417. Moran, P.A., 1950. A test for the serial dependence of residuals. Biometrika 37, 178–181. Saavedra, L.A., 2003. Tests for spatial lag dependence based on method of moments estimation. Reg. Sci. Urban Econ. 33, 27–58. Saikkonen, P., 1989. Asymptotic relative eciency of the classical test statistics under misspecification. J. Econometrics 42, 351–369. White, H., 1982. Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25. Yang, Z., 2010. A robust LM test for spatial error components. Reg. Sci. Urban Econ. 40, 299–310.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
A simple spatial dependence test robust to local and distributional misspecifications✩ Ying Fang a , Sung Y. Park b,∗ , Jinfeng Zhang c a
The Wang Yanan Institute for Studies in Economics, MOE Key Laboratory of Econometrics, Fujian Key Laboratory of Statistic Science, Xiamen University, Xiamen, Fujian 361005, China b
School of Economics, Chung-Ang University, 84 Heuksok-Ro, Dongjak-Gu, Seoul, Republic of Korea
c
The Center for Economic Research, Shandong University, Jinan, Shandong 250100, China
highlights • We propose a test for spatial dependence which is robust to local misspecification and distributional misspecification. • We find that Burridge (1980)’s test is robust to distributional misspecification. • We find that Anselin, Bera, Florax and Yoon (1996)’s test is robust to distributional misspecification.
article
info
Article history: Received 26 January 2014 Received in revised form 19 May 2014 Accepted 20 May 2014 Available online 28 May 2014
abstract In this paper, we derive a class of modified score tests robust to local and distributional misspecifications for testing spatial error autocorrelation and spatial lag dependence. The proposed tests are general enough to include several popular tests for the spatial dependence as special cases. Moreover, we show that the popular test statistics proposed by Burridge (1980) and Anselin et al. (1996) are robust to distributional misspecification although they are derived under normality assumption. © 2014 Elsevier B.V. All rights reserved.
JEL classification: C12 C21 R10 Keywords: Spatial dependence Score test Robust test Distribution misspecification Local misspecification
1. Introduction Moran (1950) proposes a seminal test, Moran’s I-test, for spatial autocorrelation in a regression model framework, but the test does not provide the nature of the spatial process that causes spatial autocorrelation, particularly, whether the spatial dependence is due to an autoregressive error process or omitted spatially lagged dependent variables. Burridge (1980) extends Moran’s I-test based on
✩ We thank the editor Badi Baltagi and one anonymous referee for their valuable comments. Fang acknowledges financial supports from NSFC 71271179, NSFC 71131008 and NCET-12-0328. Zhang acknowledges financial supports from NSFC 71301088. All remaining errors are ours. ∗ Corresponding author. Tel.: +82 2 820 5622; fax: +82 2 812 9718. E-mail address: [email protected] (S.Y. Park).
http://dx.doi.org/10.1016/j.econlet.2014.05.015 0165-1765/© 2014 Elsevier B.V. All rights reserved.
the Lagrange multiplier (LM) principle to test the spatial error autocorrelation without considering the presence of spatially lagged dependent variables, and furthermore, Anselin (1988b) proposes a LM test for the spatial error autocorrelation in the case of the presence of the spatially lagged dependent variable. However, the test involves a nonlinear optimization or applies a numerical search technique. Anselin et al. (1996) propose a modified score test for the spatial error autocorrelation in the presence of local misspecification to the parameter corresponding to the spatial lag dependence. Comparing to the Ansenlin’s LM test, the latter only requires the ordinary least squares (OLS) residuals under the null hypothesis and has little computational burden (Bera and Bilias, 2001). Yet one potential problem of these tests is that the underlying probability density may not be correctly specified, i.e., there may exist the distributional misspecification problem.
204
Y. Fang et al. / Economics Letters 124 (2014) 203–206
We develop a modified score test for spatial dependence, robust to both the local and distributional misspecifications. Local parametric misspecification arises when some nuisance parameters deviate locally from their true values. Distributional misspecification occurs when the underlying data generating process (DGP) is not correctly specified. When nuisance parameters are locally deviated from their true values, the score statistic has a non-zero drift term in general (Davidson and MacKinnon, 1987; Saikkonen, 1989). Bera and Yoon (1993) propose a modified score test robust to the local misspecification. On the other hand, when the underlying probability distribution is misspecified, some standard results are not valid any more. For example, the information matrix (IM) equality is invalid under distributional misspecification. Making inferences with the distributional misspecification can cause size distortion of the test statistics. For example, White (1982) suggests a modified LM test based on the restricted quasi-maximum likelihood (QML) estimator to deal with the distributional misspecification. Bera et al. (2007) propose a score test that is not only robust to local misspecification but also to distributional misspecification in the spirit of White (1982) and Bera and Yoon (1993). Recently, Baltagi and Yang (2013a,b) modify the standard LM test for spatial error dependence robust against distributional misspecification for cross-sectional and panel data models, respectively. In the spatial econometrics literature, the corresponding robust score (LM) test which deals with the distributional misspecification or both the local and distributional misspecifications have not been studied yet. There are some studies that suggest some other types of robust tests. However, their test statistics are based on other different estimation methods, and neither of them considers the local misspecification.1 A modified test can be constructed by properly adjusting the mean and variance of the usual score test statistics, and therefore, it has correct asymptotic size. The test statistics can be simplified when either the error term follows the normal distribution or the nuisance parameters are estimated consistently. In the next section, we first review spatial dependence tests in a spatial autoregressive model with a spatial autoregressive disturbance. We then develop new score tests robust to both local parametric and distributional misspecifications. The paper concludes in Section 3.
following Anselin (1988a), the log-likelihood function of (2.1) and (2.2) is given by N
ln 2π −
(2.1) (2.2)
where y is the dependent variable, X is a N × k matrix of explanatory variables, β denotes a k × 1 unknown parameter vector, ρ and λ are scalar spatial parameters, W1 and W2 are N × N known spatial weight matrices, ϵ is an N × 1 vector of regression disturbances, and u is an N × 1 vector of innovations with ui ∼ i.i.d(0, σ 2 ) for i = 1, 2, . . . , N. For the notational convenience, we denote θ = (β ′ , σ 2 , λ, ρ)′ , γ = (β ′ , σ 2 , λ)′ , η = (β ′ , σ 2 )′ , A = IN − ρ W1 , B = IN − λW2 , GA = W1 A−1 and GB = W2 B−1 . Under the normality assumption,
RSλ =
u˜ ′ W2 u˜ /σ˜ 2
2 ,
T22
(2.4)
where T22 = tr (W2 + W2′ )W2 , u˜ = y − X β˜ and σ˜ 2 = u˜ ′ u˜ /N.
Here θ˜ = (β˜ ′ , σ˜ 2 , 0, 0)′ denotes the constraint maximum likelihood estimator (MLE) under H0λ . The above statistic RSλ converges in distribution to χ12 under H0λ . However, if the nuisance parameter ρ ̸= 0, for example,√which is contaminated by a local deviation such that ρ = δ1 / N where δ1 is a nonzero finite constant, the Burridge statistic converges to a noncentral chi-square distribution, which implies that RSλ generally over-reject the null hypothesis even if H0λ is true. Anselin et al. (1996) construct a robust score test for the spatial error autocorrelation by eliminating the noncentral term. Using the one-step method-of-scoring estimator, the modified score test is given by
u˜ ′ W2 u˜ /σ˜ 2 − T21 N J˜ρ·η
−1
2 T22 − T21 N J˜ρ·η
u˜ ′ W1 y/σ˜ 2
2 ,
−1
(2.5)
where u˜ are the OLS residuals, σ˜ 2 = u˜ ′ u˜ /N, T21 = tr (W2 +
W2′ )W1 , J˜ρ·η =
1 N σ˜ 2
T11 σ˜ 2 +
W1 X β˜
′
MX (W1 X β) for T11 =
−1
tr (W1 + W1′ )W1 and MX = IN −X X ′ X X . RSλL converges to χ12 λ distribution under H0 even though ρ deviates locally from 0 such
√
that ρ = δ1 / N. We consider a modified score test robust to both local and distributional misspecifications, i.e., g (y) ̸= f (y, θ ) and ρ = ρ0 + √ δ1 / N where ρ0 is a finite constant and δ1 > 0. Under H0λ : λ = 0, the modified score test is given by RSλLD
−1 2 −tr(GA ) + u˜ ′ W1 y/σ˜ 2
−1
u˜ ′ W2 u˜ /σ˜ 2 − T˜2A N J˜ρ·η
= 2 N J˜ρ·η T22 − T˜2A
1 An incomplete list includes, Anselin (1990), Anselin and Kelejian (1997), Kelejian and Robinson (1998), Anselin and Moreno (2003), Saavedra (2003) and Yang (2010) among others.
1
For testing the spatial error dependence, the null hypothesis of interest is H0λ : λ = 0. Burridge (1980) proposes a one-directional score test for H0λ by assuming ρ = 0 in (2.1):
RSλL =
y = ρ W1 y + X β + ϵ, ϵ = λW2 ϵ + u,
ln σ 2 + ln |A| + ln |B| −
2.1. A modified score test for spatial error dependence
2. A modified score test robust to local and distributional misspecifications Consider the following spatial autoregressive model with a spatial autoregressive disturbance (SARAR) model (Anselin, 1988a; Case, 1991; Anselin et al., 1996; Kelejian and Prucha, 2010; among others):
N
u′ u, (2.3) 2 2 2σ 2 where u = B(Ay − X β). The test statistics previously proposed by Burridge (1980), Anselin (1988b), and Anselin et al. (1996) are derived under the normality assumption of the error terms. However, when the distribution is misspecified, all tests are generally invalid and may have the size distortion problem. Let g (y) and f (y, θ ) be the true model and the specified model, respectively. When g (y) ̸= f (y, θ ), the information matrix K (θ ) and the negative expected Hessian matrix J (θ ) are not equivalent any more; see Appendix A for details. Hence, the variance–covariance matrix of the score statistic should be modified. In general, not only the mean but also the variance of the score test statistic have to be adjusted accordingly to take care of the local and distributional misspecifications. ln L(θ ) = −
2 −2 B ˜ ∗ρ·η + N1 T˜2A J˜ρ·η
, (2.6)
where the OLS residual u˜ = y − ρ0 W1 y − X β˜ , σ˜ 2 = u˜ ′ u˜ /N and ˜ ′ T˜2A = tr (W2′ + W2 )GA , J˜ρ·η = N1 [T˜AA − N2 tr 2 (GA ) + σ12 (GA X β)
Y. Fang et al. / Economics Letters 124 (2014) 203–206
˜ , B˜ ∗ρ·η = 1 4 [2µ ˜ ′ M G¯ + κ˜ G¯ ′ G¯ ]. Here A = MX (GA X β)] ˜ 3 (GA X β) Nσ ′ X A 4 A A −1 ˜ ¯ A = v ecD (GA )− IN −ρ0 W1 , GA = W1 A , TAA = tr GA + GA GA , G ( ) µ ˜ 3 = N1 Ni=1 u˜ 3i , µ ˜ 4 = N1 Ni=1 u˜ 4i and κ˜ 4 = µ ˜ 4 − 3σ˜ 4 , a N × 1 vector of ones and v ecD (GA ) denotes a column
1 l tr GA , N N lN denotes
vector formed by the diagonal elements of GA ; see Appendix B for details.2 The modified score test statistic converges to χ12 distribution under the null hypothesis. √ Now consider a special case ρ0 = 0, i.e., ρ = δ1 / n. This yields T˜2A = T21 , tr(GA ) = 0 and B∗ρ·η = 0, which implies that Anselin et al. (1996)’s test statistics given in (2.5) is automatically robust to distributional misspecification. Moreover, when ρ = 0 in the absence of the local parametric deviation, we have Jλη = 01×(k+1) , and then Kλ = Jλ and Bλ·η = Jλ·η under H0λ . Hence, the RSλLD in (2.6) is reduced to Burridge (1980)’s test statistic in (2.4). In other words, the Burridge test statistic (RSλ ) is also robust to the distributional misspecification when ρ = 0 and λ = 0. The reason is that, even though g (y) is different from the normal distribution, ρ0 = 0 yields B∗ρ·η = 0 and tr(GA ) = 0. Therefore, the distributional misspecification does not alter the form of test statistics of Burridge (1980) and Anselin et al. (1996). Additionally, we consider the case that ρ is an unknown parameter to be estimated under the null hypothesis H0λ . Since
(β˜ ′ , σ˜ 2 , ρ) ˜ ′ is the constrained MLE under H0λ , the score term (−tr(GA ) + u˜ ′ W1 y/σ˜ 2 ) is zero. Therefore, RSλLD can be given by 2 ′ u˜ W2 u˜ /σ˜ 2 LD RS λ = (2.7) −1 . 2 2 −2 B ˜ ∗ρ·η T22 − T˜2A N J˜ρ·η + N1 T˜2A J˜ρ·η Note that Var (ρ) ˜ = N J˜ρ·η in this case. Compared to Anselin
(1988b), the only difference comes from the fact that we correct the variance of the score test statistic in (2.7) to take care of the distributional misspecification. We conduct Monte Carlo simulations to examine the finite sample performance of the proposed tests.3 For the data generating process, we consider four different distributions of u in (2.2), normal, Student’s t, gamma and asymmetric bimodal mixture normal distributions, and three choices of weight matrices including the queen contiguity weight, the circular weight and the Anselin weight. We compare the size performances of RSλ and RSλLD under different distributions and different values of ρ . Note that in this case our proposed test RSλLD is identical to RSλL . The empirical sizes of both tests are quite correct when there exists only distributional misspecification. This is due to the robustness of both statistics. However, the actual sizes of RSλ increase quickly in the presence of local misspecification, while the size performance of RSλLD is relatively stable. We also compare
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2.2. A modified score test for spatial autoregressive dependence A modified score test robust to local and distributional misspecifications for testing spatial autoregressive dependence can be derived as in (2.6), and the detail is given in Appendix C.4 Under ρ the null hypothesis H0 : ρ = 0, the modified score test is written by RSρLD
=
u˜ ′ BW1 y/σ˜ 2 − T˜BC1 (N J˜ρ·η )−1 −tr(GB ) + u˜ ′ GB u˜ /σ˜ 2
N J˜ρ·η −
˜
1 2 T N BC1
2
(J˜λ·η )−1 + N B˜ ∗ρ·η + C˜ ρ∗
, (2.8)
˜ and σ˜ 2 = u˜ ′ u˜ /N. T˜BC1 = where u˜ = (I − λ0 W2 )(y − X β) tr (G′B + GB )C1 , J˜λ·η = N12 NTBB − 2tr 2 (GB ) , J˜ρ·η = N1 [TC1 C1 + 1 ˜ ′ MBX (BW1 X β)] ˜ , TC1 C1 = tr (C1′ + C1 )C1 , C1 = (BW1 X β) σ˜ 2
BW1 B−1 , B˜ ∗ρ·η + C˜ ρ∗ = B = IN − λ0 W2 , GB J˜ρλ·η =
1 T N BC1
′ ˜ ′ MBX F˜ . Here κ˜ 4 F˜ F˜ + 2µ ˜ 3 (BW1 X β) −1 ˜¯ = W2 B−1 , F˜ = v ec D (C1 ) − J˜ρλ·η J˜λ·η GB , 1 N σ˜ 4
˜¯ = v ec (G ) − and G B D B
(GB ).
1 l tr N N
Similar to the case of RSλLD , i.e., λ = 0 in (2.2) the RSρLD in (2.8) is reduced to Anselin (1988b)’s equation (32) when λ is absent in the model. When λ0 = 0, the drift term in (2.8) is the same as that in Anselin et al. (1996). Moreover, since F˜ = 0 yields B˜ ∗ρ·η + C˜ ρ∗ = 0 in this case, the denominator in (2.8) is also the same as the denominator of equation 14 of Anselin et al. (1996), which implies equation 14 of Anselin et al. (1996), is automatically robust to the distributional misspecification when λ0 = 0. We also consider the case that λ is a parameter to be estimated ρ ˜ ′ is the constrained MLE under the null under H0 . Since (β˜ ′ , σ˜ 2 , λ) ρ
H0 , the score term S˜λ1 = 0. Therefore, the RSρLD is given by u˜ ′ BW1 y/σ˜ 2
LD
RS ρ =
N J˜ρ·η −
˜
1 2 T N BC1
2
. (J˜λ·η )−1 + N B˜ ∗ρ·η + C˜ ρ∗
(2.9)
Compared to Anselin (1988b), we adjust only the variance of the score test in (2.9) to correct the distributional misspecification. We also perform the similar Monte-Carlo simulations to anaLD
lyze the empirical sizes and powers of RSρ , RSρL , RSρLD and RS ρ using
the same simulation designs for the spatial error dependence case.5 The simulation results are quite similar with those of the spatial error dependence case. 3. Conclusion
LD
Anselin (1988b)’s test to our proposed test RS λ in the case that ρ is unknown and ρ0 ̸= 0. Since the nuisance parameter for both tests needs to be estimated, the actual sizes of both tests are robust to local misspecification. However, the impact of distributional misspecification on empirical sizes seems to be small. Finally, the power performance of all tests looks very good for all experiment designs. Our proposed test RSλLD achieves considerable powers when the model is either locally or globally misspecified. Moreover, RSλLD is more powerful than Anselin (1988b)’s test when there exist local and distributional misspecifications.
2 The Appendix can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf. 3 To conserve space we do not report the Monte-Carlo simulation designs and results but these can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf.
In this paper we derive modified score tests robust to both local and distributional misspecifications in the spatial autoregressive model with a spatial autoregressive error term. We show that some popular spatial dependence tests, such as Burridge (1980) and Anselin et al. (1996), can be expressed as special cases of our tests. We also find that Burridge (1980) and Anselin et al. (1996)’s tests are automatically robust to distributional misspecification under some special cases. Our findings in this paper support the usage of Anselin et al. (1996)’s tests for spatial dependence since they are also robust to distributional misspecification.
4 The Appendix can be obtained from the web page: www.sungpark.net/FPZ_EL_ MC_Tables_Appendix.pdf. 5 The test statistics, RS and RS L , are similar to RS and RS L , respectively, and can ρ
ρ
λ
be found in Burridge (1980) and Anselin et al. (1996).
λ
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Y. Fang et al. / Economics Letters 124 (2014) 203–206
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