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Locally adjusted LM test for spatial dependence in fixed effects panel data models
Economics Letters 121 (2013) 59–63
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Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Locally adjusted LM test for spatial dependence in fixed effects panel data models Ming He a,∗,1 , Kuan-Pin Lin b,1 a
Department of Economics, Vanderbilt University, VU Station B #351819, 2301 Vanderbilt Place, Nashville, TN 37235-1819, United States
b
Department of Economics, Portland State University, 1721 SW Broadway, Cramer Hall, Suite 241, Portland, OR 97201, United States
highlights • We adopt the spatial panel one-way and two-way fixed effects models. • We construct robust LM tests to guard against local misspecification. • We conduct simulation to show performance of robust LM tests.
article
info
Article history: Received 2 April 2013 Received in revised form 24 June 2013 Accepted 28 June 2013 Available online 11 July 2013 Keywords: Spatial dependence Locally adjusted (robust) LM test Bera and Yoon principle Fixed effects panel data model
abstract Lee and Yu (2010) propose spatial panel data models with one-way and two-way fixed effects. Debarsy and Ertur (2010) construct LM (Lagrange multiplier) and LR (likelihood ratio) tests in the one-way fixed effects model. He and Lin (2012) derive LM tests in the two-way fixed effects model. To guard against possible local misspecification, in this paper we apply Bera and Yoon (1993) principle, and construct locally adjusted (robust) LM tests for spatial dependence in both one-way and two-way fixed effects models. Monte Carlo experiment is carried out to show the advantage of using robust LM tests over the corresponding marginal and conditional versions. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Spatial econometric models have been extensively studied in the past two decades. They are different from the traditional models in the sense that they consider spatial correlation of different observations instead of viewing them as independent. Ignoring spatial effects when they are actually present leads, at best, to inefficient ordinary least squares (OLS) estimators and at worst, to biased and inconsistent OLS estimators. There are two popular forms of spatial correlation. The first is the spatial autoregressive (SAR) process which assumes correlation of dependent variables among neighbors. The second is the spatial error autocorrelation which allows the unobserved error terms to be spatially correlated. Hypothesis testing for spatial dependence has been studied extensively in the literature. For tests of spatial dependence in cross section models, see Anselin (1988a,b, 2001), Anselin et al. (1996),
∗
Corresponding author. Tel.: +1 503 863 9798. E-mail addresses: [email protected] (M. He), [email protected] (K.-P. Lin).
1 The authors would like to thank the anonymous referee for his (her) comments which have greatly helped to improve the paper. 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.06.039
Anselin and Bera (1998) and Qu and Lee (2012). For tests of spatial dependence in panel data models, see Baltagi et al. (2003, 2007, 2013), Baltagi and Liu (2008), Debarsy and Ertur (2010) and He and Lin (2012). Many of the above tests are based on the LM principle. Bera and Yoon (1993) argue that the LM test corresponding to marginal hypothesis might suffer from size distortion under local misspecification of some nuisance parameters, while the LM test corresponding to conditional hypothesis requires maximum likelihood estimation which might be computationally cumbersome. They develop locally adjusted (robust) LM test to guard against local deviation of nuisance parameters from certain pre-specified values. We emphasize two properties of the robust LM test. First, if the null hypothesis is true while the nuisance parameters deviate from prespecified values, the marginal LM test will follow a non-centralized χ 2 distribution asymptotically, tending to reject the null hypothesis too often. However, the robust LM test in this case will follow a centralized χ 2 distribution, thus providing correct asymptotic size, as long as the deviation is local. Second, if the null hypothesis is not true while the nuisance parameters actually take on the pre-specified values, the robust LM test tends to accept the null more often than the marginal LM test, thus it will be less powerful
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M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
than the marginal LM test. Moreover, robust LM test only requires computing the restricted estimator under relevant joint null hypothesis, and it is the OLS estimator in most cases. Therefore, it is easier to compute than the corresponding conditional LM test. In practice, the robust LM test should be used together with marginal and conditional LM tests in order to make correct statistical inference. In this paper, we extend the works in Debarsy and Ertur (2010) and He and Lin (2012) by deriving robust LM tests for the one-way and two-way fixed effects spatial panel data models developed in Lee and Yu (2010). Since a certain condition (see Section 3.1 below) for the marginal and robust LM tests being equal is not satisfied, then the marginal LM tests constructed in Debarsy and Ertur (2010) and He and Lin (2012) cannot be used to guard against local misspecification. Instead, the robust LM tests should be used. The rest of the paper proceeds as follows: model specifications are briefly reviewed in Section 2. In Section 3, four robust LM tests are provided. In Section 4, we report Monte Carlo experiment results to demonstrate finite sample performance of these robust LM tests. We show the advantage of using robust LM tests by comparing them to the corresponding marginal and conditional LM tests. A conclusion is made in Section 5. 2. Model specification
∗ where Sn (λ) = In −λWn , Rn (ρ) = In −ρ Mn , Vnt = Rn (ρ)[Sn (λ)Ynt∗ − ∗ Xnt β]. √ Similarly, in the two-way fixed effects model, let [Fn,n−1 , ln / n] be the orthonormal eigenvector matrix of the demeaning operator Jn = In − ln l′n /n, where ln is a n × 1 vector of ones. Fn,n−1 is the n × (n − 1) submatrix corresponding to the eigenvalues one. ∗∗ Let Ynt = Fn′ ,n−1 Ynt∗ , Xnt∗∗ = Fn′ ,n−1 Xnt∗ , t = 1, . . . , T . After transformation, the log-likelihood function of the two-way fixed effects model is (n − 1)(T − 1) ln(2π σ 2 ) L2 (θ) = − 2 + (T − 1)[ln |Sn (λ)| + ln |Rn (ρ)|]
− (T − 1)[ln(1 − λ) + ln(1 − ρ)] −
T −1 1 ∗ ∗′ ∗∗ Vnt Vnt 2
2σ
(4)
t =1
∗∗ where Vnt = R∗n (ρ)[Sn∗ (λ)Ynt∗∗ − Xnt∗∗ β], R∗n (ρ) = In−1 − ρ Mn∗ , Sn∗ (λ) = In−1 − λWn∗ , Mn∗ = Fn′ ,n−1 Mn Fn,n−1 , Wn∗ = Fn′ ,n−1 Wn Fn,n−1 . Lee and Yu (2010) show that the maximum likelihood estimators based on the log-likelihood functions (3) and (4), under a set of regularity assumptions, are consistent and centered at the true values of population parameters.
In this paper, we consider fixed effects panel data models with first order SAR lag and first order SAR disturbances. Using the notations in Lee and Yu (2010), the SAR panel data model with individual effects (one-way fixed effects) and SAR disturbances is
3. Bera and Yoon principle, robust LM tests
Ynt = λ0 Wn Ynt + Xnt β0 + cn0 + Unt
Bera and Yoon (1993) discuss the issue of local misspecification regarding to some nuisance parameters. Let θ = (γ ′ , ψ ′ , φ ′ )′ and the log-likelihood function be L(θ), and denote γ as the restricted estimator of γ under the joint null hypothesis H0 : ψ = ψ0 , φ = φ0 . Let θ = ( γ ′ , ψ0′ , φ0′ )′ and Iψγ ′ = −E [∂ 2 L(θ)/∂ψ∂γ ′ ], etc. The robust LM test statistic corresponding to the marginal null hypothesis H0 : ψ = ψ0 (φ = φ0 ) is given by
Unt = ρ0 Mn Unt + Vnt
(1)
t = 1, 2, . . . , T
while the SAR panel data model with both individual and time effects (two-way fixed effects) and SAR disturbances is Ynt = λ0 Wn Ynt + Xnt β0 + cn0 + αt0 ln + Unt Unt = ρ0 Mn Unt + Vnt
(2)
t = 1, 2, . . . , T
where Ynt = (y1t , y2t , . . . , ynt ) and Vnt = (v1t , v2t , . . . , vnt ) are n × 1 vectors for period t, vit is i.i.d. across i and t with zero mean and variance σ02 . Xnt is a n × k matrix of time varying explanatory variables for period t. cn0 is a n × 1 vector representing the individual fixed effects, while αt0 represents the time effects in period t. Wn , Mn are exogenous spatial weights matrices associated with spatially correlated dependent variable and spatially correlated error. Lee and Yu (2010) show that the maximum likelihood estimator (MLE) based on a direct approach (i.e. MLE from the above two models) does not consistently estimate σ02 unless T is large. To solve this problem, they propose a certain transformation approach to eliminate the√ fixed individual and time effects. Specifically, let [FT ,T −1 , lT / T ] be the orthonormal eigenvector matrix of the demeaning operator JT = IT − lT l′T /T , where IT is an identity matrix of size T , and lT is a T × 1 vector of ones. FT ,T −1 is the T × (T − 1) submatrix corresponding to the eigenvalues ′ ′ ′ ′ ∗′ , . . . , YnT ) , where ⊗ one. Let (Yn1 , . . . , Yn∗,T −1 )′ = (FT′ ,T −1 ⊗ In )(Yn1 denotes the Kronecker product, and In is an identity matrix of size ∗ n. Xnt , t = 1, . . . , T are defined similarly (see Lee and Yu (2010) for more details of the transformation approach and the regularity assumptions). Let θ = (β′ , λ, ρ, σ 2 )′ , then after transformation, the loglikelihood function of the one-way fixed effects model is ′
L1 (θ) = −
n(T − 1) 2
′
T −1 1 ∗′ ∗ Vnt Vnt 2
2σ
t =1
LM∗ψ =
∂ L∗ ∂ψ
′
(I∗ )−1
∂ L∗ ∂ψ
θ
−1 ∗ −1 where (∂ L∗ /∂ψ)|θ = (∂ L/∂ψ − Iψφ ′ ·γ Iφφ ′ ·γ ∂ L/∂φ)| θ , (I ) | θ =
−1 −1 −1 (Iψψ ′ ·γ − Iψφ ′ ·γ Iφφ |θ and Iψψ ′ ·γ = Iψψ ′ − Iψγ ′ Iγγ ′ ·γ Iφψ ′ ·γ ) ′ −1 ′ ′ ′ ′ ′ ′ ′ ′ Iγψ , Iφφ ·γ = Iφφ − Iφγ Iγγ ′ Iγφ , Iψφ ·γ = Iφψ ′ ·γ = Iψφ −
−1 Iψγ ′ Iγγ ′ Iγφ ′ . The crucial quantity in the above formula is Iψφ ′ ·γ . If Iψφ ′ ·γ = 0, then LM∗ψ = LMψ , where LMψ denotes the marginal
LM test for ψ . Bera and Yoon (1993) also show the identity that LMψφ = LMφ + LM∗ψ = LMψ + LM∗φ , where LMψφ , LMφ , LM∗φ denote the joint LM test for ψ, φ , the marginal and robust LM tests for φ , respectively. Thus if Iψφ ′ ·γ = 0, LMψφ = LMφ + LMψ . The advantage of the robust LM test is that it only requires estimation under the joint null hypothesis, thus is simpler, and it can provide valid asymptotic size when the nuisance parameters deviate locally from the pre-specified values (for example, φ deviates locally from φ0 in the case of marginal test for ψ ). However, when indeed φ = φ0 in the population model, LM∗ψ will generally be less powerful than LMψ . 3.2. Robust LM tests in the one-way fixed effects model In this section, we consider two robust LM tests in the one-way fixed effects model, two corresponding hypotheses are considered. Hypothesis (1.1): H01.1 : λ = 0 (allowing for local deviation of ρ from 0) vs. H11.1 : otherwise.
ln(2π σ 2 ) + (T − 1)[ln |Sn (λ)|
+ ln |Rn (ρ)|] −
3.1. Bera and Yoon principle
(3)
Hypothesis (1.2): H01.2 : ρ = 0 (allowing for local deviation of λ from 0) vs. H11.2 : otherwise.
M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
The corresponding joint null hypothesis is H0 : ρ = λ = 0. The robust LM tests are given by3
z1λ − LM∗1λ = ξ1λ LM∗1ρ
= ξ1ρ z1ρ −
b12 b13
2 z1ρ
, 2
b12 b11 + ω1
z1λ
where ξ1λ = b13 /(b11 b13 − b212 + b13 ω1 ), ξ1ρ = (b11 + ω1 )/
∗′ ∗ (b11 b13 − b212 + b13 ω1 ), z1λ = (1/ σ12 ) Tt =−11 Vnt Wn Ynt , z1ρ = (1/ σ12 ) T −1 ∗′ ∗ ′ t =1 Vnt Mn Vnt , b11 = (T − 1)tr[(Wn + Wn )Wn ], b12 = (T − 1)tr[(Mn + Mn′ )Wn ], b13 = (T − 1)tr[(Mn + Mn′ )Mn ], ω = (1/ σ 2) T −11 ∗′ ∗ 1−1 T −1 ∗′ T −1 2 ∗ ′ ∗ ∗ ′ Ynt ) ( t =1 Xnt Xnt ) Y )−(1/ σ1 )( t =1 Xnt Wn Y ) (W (W t =1 n′ nt ∗ n nt ∗ ∗ ( tT=−11 Xnt∗ Wn Ynt ), and Vnt = Ynt∗ − Xnt∗ β1 , Ynt = Xnt∗ β1 , β1 = T − 1 ∗′ ∗ − 1 T − 1 ∗′ ∗ ′ T − 1 2 ∗ ∗ ( t =1 Xnt Xnt ) ( t =1 Xnt Ynt ), σ1 = t =1 Vnt Vnt /[n(T − 1)]. ′ ′ Let θ 1 = ( β 1 , 0, 0, σ12 )′ , then γ 1 = ( β1 , σ12 )′ . In hypothesis (1.1), ψ = λ, φ = ρ , and in hypothesis (1.2), ψ = ρ, φ = λ. It can be shown that (I1λρ·γ )|θ 1 = (I1ρλ·γ )|θ 1 = b12 ̸= 0, then ∂ L1 / ∂λ and ∂ L1 /∂ρ are partially correlated even after eliminating the effects of ∂ L1 /∂γ on ∂ L1 /∂λ and ∂ L1 /∂ρ , rendering the differences between the robust LM tests and the marginal LM tests, which are given by LM1λ =
1 b11 + ω1
z12λ ,
LM1ρ =
1 b13
z12ρ .
On one hand, if the data is generated according to λ = 0 (ρ ̸= 0), the asymptotic size of LM1λ will be distorted, whereas LM∗1λ converges to a centralized χ12 distribution under local deviation of ρ from 0. In this situation, LM∗1λ should be used for statistical inference. On the other hand, if ρ = 0 in the population model, LM∗1λ will generally be less powerful than LM1λ . A similar discussion applies to LM∗1ρ and LM1ρ .
between the robust LM tests and the marginal LM tests, which are given by LM2λ =
1 b21 + ω2 ∗
In this section, we consider two robust LM tests in the two-way fixed effects model, two corresponding hypotheses are considered. Hypothesis (2.1): H02.1 : λ = 0 (allowing for local deviation of ρ from 0) vs. H12.1 : otherwise. Hypothesis (2.2): H02.2 : ρ = 0 (allowing for local deviation of λ from 0) vs. H12.2 : otherwise. The corresponding joint null hypothesis is H0 : ρ = λ = 0. The robust LM tests are given by b∗ z2ρ LM∗2λ = ξ2λ z2λ − ∗22 b23
2
,
b∗ LM∗2ρ = ξ2ρ z2ρ − ∗ 22 z2λ ω2 b21 +
2
ω2 )/ ω2 ), ξ2ρ = (b∗21 + where ξ2λ = b∗23 /(b∗21 b∗23 − b∗222 + b∗23
∗ ∗′ Wn∗ ω2 ), z2λ = (T − 1) + (1/ σ22 ) Tt =−11 Vnt (b∗21 b∗23 − b∗222 + b∗23 ′ T −1 ∗ ∗ 2 ∗∗∗ ∗ ∗∗ Ynt , z2ρ = (T − 1) + (1/ σ2 ) t =1 Vnt Mn Vnt , b2j = b2j − 2(T − ′ 1)/(n − 1), j = 1, 2, 3, b21 = (T − 1)tr[(Wn∗ + Wn∗ )Wn∗ ], b22 = ′ ′ (T − 1)tr[(Mn∗ + Mn∗ )Wn∗ ], b23 = (T − 1)tr[(Mn∗ + Mn∗ )Mn∗ ], and V ∗∗ T −1 ∗ ∗∗ ′ nt∗ 2 ∗∗ ∗∗ ∗∗ ∗∗ = Ynt − Xnt β2 , Ynt = Xnt β2 , ω2 = (1/ σ2 ) t =1 (Wn Ynt ) (Wn ′ ′ ′ ∗∗ ∗∗ ′ Ynt ) − (1/ σ22 )( Tt =−11 Xnt∗ ∗ Wn∗ Ynt ) ( Tt =−11 Xnt∗ ∗ Xnt∗∗ )( Tt =−11 Xnt∗ ∗ Wn∗
3 We do not present the derivation of test statistics in this paper, but it can be found in the earlier manuscript He and Lin (2013).
z22λ ,
LM2ρ =
1 b∗23
z22ρ .
On one hand, if the data is generated according to λ = 0 (ρ ̸= 0), the asymptotic size of LM2λ will be distorted, whereas LM∗2λ converges to a centralized χ12 distribution under local deviation of ρ from 0. In this situation, LM∗2λ should be used for statistical inference. On the other hand, if ρ = 0 in the population model, LM∗2λ will generally be less powerful than LM2λ . A similar discussion applies to LM∗2ρ and LM2ρ . 4. Monte Carlo simulations In this section, we conduct a small Monte Carlo simulation to evaluate the size and power performance of the proposed test statistics. The data generating process is, for t = 1, . . . , T , Ynt = λ0 Wn Ynt + Xnt1 β10 + Xnt2 β20
+ cn0 + αt0 ln + (In − ρ0 Mn )−1 Vnt the time effects term—αt0 is suppressed in the one-way fixed effects model. Vnt = (v1t , v2t , . . . , vnt )′ , vit ’s are i.i.d. across i and t. We generate vit from a mixed distribution, with probability 0.5 from a log-normal distribution, and with probability 0.5 from a Weibull distribution. Specifically,
vit = rit − E [rit ], 3.3. Robust LM tests in the two-way fixed effects model
61
′ ′ ∗∗ ∗ ∗′ Ynt ), β2 = ( tT=−11 Xnt∗ ∗ Xnt∗∗ )−1 ( Tt =−11 Xnt∗ ∗ Ynt∗∗ ), σ22 = Tt =−11 Vnt ∗∗ Vnt /[(n − 1)(T − 1)]. ′ ′ Let θ 2 = ( β 2 , 0, 0, σ22 )′ , then γ 2 = ( β2 , σ22 )′ . In hypothesis (2.1), ψ = λ, φ = ρ , and in hypothesis (2.2), ψ = ρ, φ = λ. It can be shown that (I2λρ·γ )|θ 2 = (I2ρλ·γ )|θ 2 = b∗22 ̸= 0, then ∂ L2 /∂λ and ∂ L2 /∂ρ are partially correlated even after eliminating the effects of ∂ L2 /∂γ on ∂ L2 /∂λ and ∂ L2 /∂ρ , rendering the differences
rit = (1 − Wit )Z1,it + Wit Z2,it ,
where Wit is a Bernoulli random variable with success probability p = 0.5, log Z1,it ∼ N (−2, 4), Z2,it is generated from a Weibull distribution with scale parameter 5 and shape parameter 1, Wit , Z1,it , Z2,it are independent of each other. rit is demeaned by the population mean so that E [vit ] = 0. For the individual effects, each element of cn0 is generated independently from a uniform distribution on [−5, 5]. For the time effects, α(t +1)0 = 1.05αt0 , α10 is generated from a uniform distribution on [0, 10]. Each element of Xnt1 is generated independently from N (0, 16), while each element of Xnt2 is generated independently from a uniform distribution on [0, 10]. For the spatial weights matrices, Wn = Mn , and they both take the form of a first order rook contiguity matrix. For the parameter values, we set β10 = 0.5, β20 = 0.7, both λ0 and ρ0 take values in [−0.8, 0.8], with increment 0.2. For each combination of parameter values, two sample sizes are chosen, namely, n = 81, T = 8, and n = 196, T = 10, with 1000 replications performed. The nominal size is set to be 0.05. The empirical size and power of the marginal and robust LM tests in the one-way fixed effects model is summarized in Table 1. We compare the empirical size of the marginal and robust LM tests under local misspecification, and their empirical power without local misspecification. For the robust test LM∗1λ , when λ = 0 (ρ ̸= 0), it follows a centralized χ 2 distribution asymptotically. It shows strength in correctly accepting the null hypothesis λ = 0, while LM1λ incorrectly reject the null hypothesis. For example, when the sample size is (81, 8) and λ = 0 (ρ = 0.4), the frequency of rejection (FR) of LM∗1λ is 0.065, while the FR of LM1λ is 1. Generally, the empirical size of LM∗1λ is reasonable when |ρ| ≤ 0.4, this is because we only allow for local deviation of ρ from 0. The size performance of LM∗1λ is similar for the (196, 10) sample. Therefore, when
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M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
Table 1 Size and power comparison of marginal and robust LM tests, one-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ=0 ρ
Size LM1λ
LM∗1λ
LM1ρ
LM∗1ρ
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.917 0.029 0.925 1.000 1.000 1.000
0.195 0.111 0.068 0.064 0.036 0.065 0.065 0.101 0.136
1.000 1.000 1.000 0.955 0.033 0.963 1.000 1.000 1.000
0.952 0.918 0.697 0.264 0.039 0.252 0.733 0.913 0.951
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 1.000 0.046 1.000 1.000 1.000 1.000
0.161 0.117 0.071 0.051 0.049 0.053 0.075 0.088 0.154
1.000 1.000 1.000 1.000 0.041 1.000 1.000 1.000 1.000
0.992 0.989 0.956 0.595 0.042 0.569 0.953 0.989 0.993
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
Power
ρ=0 λ
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Size
Power
Table 2 Size and power comparison of robust and conditional LM tests, one-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ
LM1ρ
LM∗1ρ
LM1λ
LM∗1λ
1.000 1.000 1.000 0.941 0.033 0.968 1.000 1.000 1.000
0.035 0.050 0.042 0.045 0.039 0.053 0.049 0.041 0.030
1.000 1.000 1.000 0.968 0.029 0.983 1.000 1.000 1.000
0.979 0.963 0.806 0.322 0.036 0.325 0.802 0.946 0.976
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.999 0.041 1.000 1.000 1.000 1.000
0.046 0.057 0.059 0.055 0.042 0.041 0.055 0.052 0.035
1.000 1.000 1.000 0.999 0.046 1.000 1.000 1.000 1.000
0.994 0.991 0.965 0.624 0.049 0.631 0.961 0.993 0.996
−0.8 −0.6 −0.4 −0.2
λ = 0 (ρ ̸= 0), using marginal test LM1λ will be misleading. However, there is one drawback of applying the robust LM test, that is, when the assumption of the nuisance parameter (ρ = 0 in this case) is indeed true, LM∗1λ is less powerful than the marginal test LM1λ . For instance, when λ = −0.4 (ρ = 0) and the sample size is (81, 8), the FR of LM1λ is 1, while that of LM∗1λ is 0.806. However, the loss of power is not very severe, and it improves as n becomes larger. For example, when λ = −0.4 (ρ = 0), the FR of LM∗1λ is 0.965 for the (196, 10) sample. The performance of LM∗1ρ follows a similar discussion as that of LM∗1λ , and its size performance is even better in the sense that it provides correct asymptotic size even when λ deviates far away from 0. In the one-way fixed effects model, we also compare performances of the robust LM tests with those of the corresponding conditional LM tests. The results are summarized in Table 2. For the sake of compactness, we only report the results when the nuisance parameters take values −0.4, 0.4. LM1λ|ρ and LM1ρ|λ are the conditional LM tests in Debarsy and Ertur (2010). In the left four columns, generally, the size and power performances of LM∗1λ and LM1λ|ρ are comparable. For example, when the sample size is (81, 8) and ρ = 0.4, the empirical size of LM∗1λ is 0.065, while that of LM1λ|ρ is 0.060. Both LM∗1λ and LM1λ|ρ are a little oversized for the (196, 10) sample. For instance, when ρ = −0.4, the empirical size of LM∗1λ is 0.071, and that of LM1λ|ρ is 0.080. There is an interesting pattern in their power performances. When ρ = −0.4, the power of LM∗1λ is lower than that of LM1λ|ρ when λ < 0, while the reverse order holds when λ > 0. And there is a similar but reverse pattern when ρ = 0.4. Also, the FR of LM1λ|ρ becomes very low when λ = 0.8 (ρ = −0.4) and λ = −0.8 (ρ = 0.4), although this loss in power improves for the (196, 10) sample. A similar discussion applies to the comparison between LM∗1ρ and LM1ρ|λ . The performances and comparison of tests in the two-way fixed effects model are similar to those in the one-way fixed effects model, and they are summarized in Tables 3 and 4. In general, the robust tests LM∗1λ , LM∗1ρ , LM∗2λ , LM∗2ρ have very good performances in correctly accepting the null hypothesis when the nuisance parameters deviate from pre-specified values. Although there is a loss in power compared to the marginal LM tests when the assumption of the nuisance parameters is true, the loss is not very severe, and it improves as the sample size becomes larger. Also, when the nuisance parameters deviate from pre-specified values, performances of the robust LM tests are generally comparable to those of the conditional LM tests. Finally, the robust LM tests are computationally easier to obtain than the
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
ρ = −0.4
ρ = 0.4
LM∗1λ
LM1λ|ρ
LM∗1λ
LM1λ|ρ
ρ
0.938 0.848 0.684 0.302 0.068 0.318 0.805 0.970 0.984
0.982 0.974 0.797 0.355 0.059 0.356 0.793 0.650 0.248
0.993 0.971 0.820 0.307 0.065 0.284 0.662 0.897 0.933
0.362 0.654 0.796 0.391 0.060 0.371 0.830 0.971 0.978
−0.8 −0.6 −0.4 −0.2
0.992 0.972 0.909 0.549 0.071 0.582 0.958 0.985 0.997
1.000 0.996 0.971 0.731 0.080 0.746 0.968 0.873 0.495
0.998 0.987 0.971 0.589 0.075 0.530 0.917 0.980 0.989
0.608 0.829 0.965 0.751 0.068 0.689 0.973 0.997 1.000
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
λ = −0.4
λ = 0.4
LM∗1ρ
LM1ρ|λ
LM∗1ρ
LM1ρ|λ
0.747 0.759 0.540 0.199 0.042 0.264 0.769 0.951 0.981
0.991 0.966 0.792 0.319 0.061 0.373 0.803 0.806 0.261
0.976 0.955 0.745 0.236 0.049 0.205 0.546 0.763 0.760
0.306 0.812 0.792 0.379 0.049 0.319 0.779 0.953 0.993
0.956 0.971 0.900 0.467 0.059 0.542 0.966 0.992 0.997
1.000 0.997 0.972 0.688 0.077 0.720 0.960 0.926 0.400
0.991 0.992 0.957 0.523 0.055 0.508 0.909 0.973 0.974
0.386 0.913 0.968 0.714 0.085 0.677 0.972 0.997 1.000
Table 3 Size and power comparison of marginal and robust LM Tests, two-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ=0 ρ
Size LM2λ
LM∗2λ
LM2ρ
LM∗2ρ
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.916 0.034 0.899 1.000 1.000 1.000
0.194 0.112 0.065 0.059 0.034 0.059 0.063 0.088 0.121
1.000 1.000 1.000 0.952 0.036 0.946 1.000 1.000 1.000
0.951 0.907 0.685 0.267 0.039 0.226 0.711 0.908 0.953
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 1.000 0.046 1.000 1.000 1.000 1.000
0.165 0.119 0.074 0.049 0.048 0.059 0.070 0.096 0.144
1.000 1.000 1.000 1.000 0.043 1.000 1.000 1.000 1.000
0.992 0.989 0.953 0.595 0.050 0.567 0.956 0.989 0.994
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
Power
ρ=0 λ
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Size
Power
LM2ρ
LM∗2ρ
LM2λ
LM∗2λ
1.000 1.000 1.000 0.934 0.036 0.965 1.000 1.000 1.000
0.037 0.046 0.043 0.048 0.039 0.044 0.046 0.039 0.027
1.000 1.000 1.000 0.965 0.034 0.981 1.000 1.000 1.000
0.978 0.960 0.805 0.316 0.034 0.318 0.780 0.939 0.971
1.000 1.000 1.000 0.999 0.043 1.000 1.000 1.000 1.000
0.047 0.055 0.060 0.053 0.050 0.039 0.045 0.050 0.038
1.000 1.000 1.000 0.999 0.046 1.000 1.000 1.000 1.000
0.994 0.989 0.965 0.622 0.048 0.626 0.962 0.993 0.994
Table 4 Size and power comparison of robust and conditional LM tests, two-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
ρ = −0.4
ρ = 0.4
LM∗2λ
LM2λ|ρ
LM∗2λ
LM2λ|ρ
ρ
0.937 0.842 0.668 0.301 0.065 0.309 0.781 0.965 0.982
0.982 0.972 0.779 0.333 0.063 0.343 0.764 0.667 0.187
0.992 0.971 0.818 0.307 0.063 0.273 0.649 0.894 0.934
0.320 0.648 0.785 0.368 0.065 0.351 0.819 0.962 0.954
−0.8 −0.6 −0.4 −0.2
0.992 0.971 0.907 0.533 0.074 0.575 0.955 0.985 0.997
1.000 0.997 0.972 0.711 0.079 0.723 0.966 0.877 0.416
0.998 0.986 0.970 0.583 0.070 0.511 0.901 0.977 0.988
0.581 0.829 0.963 0.746 0.071 0.687 0.969 0.997 1.000
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
λ = −0.4
λ = 0.4
LM∗2ρ
LM2ρ|λ
LM∗2ρ
LM2ρ|λ
0.736 0.758 0.546 0.198 0.043 0.249 0.754 0.945 0.980
0.992 0.964 0.772 0.316 0.064 0.358 0.792 0.812 0.347
0.972 0.944 0.728 0.235 0.046 0.207 0.533 0.760 0.801
0.285 0.799 0.765 0.379 0.055 0.306 0.770 0.940 0.993
0.956 0.970 0.899 0.461 0.060 0.536 0.963 0.991 0.997
1.000 0.996 0.973 0.675 0.073 0.716 0.965 0.931 0.441
0.991 0.994 0.951 0.515 0.045 0.478 0.908 0.973 0.976
0.379 0.909 0.966 0.702 0.085 0.675 0.969 0.997 1.000
M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
conditional LM tests since they only require OLS estimation, which greatly saves the computation time. For instance, when the sample size is (196, 10), the simulation time of LM∗1λ is 826 s in the oneway effects model, while that of LM1λ|ρ is 14 859 s, which is almost 18 times larger than that of LM∗1λ . In sum, researchers are recommended to use the robust LM tests together with the marginal and conditional LM tests to draw correct statistical inference. 5. Conclusion In this paper, we adopt the spatial panel data models with oneway and two-way fixed effects in Lee and Yu (2010). We construct the robust LM test statistics based on the Bera and Yoon (1993) principle. To complement the works in Debarsy and Ertur (2010) and He and Lin (2012), we provide the robust LM tests for a spatially correlated dependent variable allowing for the local presence of spatially correlated error and the robust LM tests for spatially correlated error allowing for the local presence of a spatially correlated dependent variable. Monte Carlo simulations show the advantage of using robust LM tests over the corresponding marginal and conditional LM tests. In practice, researchers are suggested to use robust LM tests together with marginal and conditional LM tests to draw correct statistical inference. Finally, in our Monte Carlo study, we find that the performance of the robust LM tests for a spatially correlated error is generally better than that of the robust LM tests for a spatially correlated dependent variable.
63
References Anselin, L., 1988a. Lagrange multiplier test diagnostics for spatial dependence and spatial heterogeneity. Geographical Analysis 20, 1–17. Anselin, L., 1988b. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., 2001. Rao’s score tests in spatial econometrics. Journal of Statistical Planning and Inference 97, 113–139. Anselin, L., Bera, A.K., 1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York. Anselin, L., Bera, A.K., Florax, R., Yoon, M., 1996. Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics 26, 77–104. Baltagi, B.H., Egger, P., Pfaffermayr, M., 2013. A generalized spatial panel data model with random effects. Econometric Reviews 32, 650–685. Baltagi, B.H., Liu, L., 2008. Testing for random effects and spatial lag dependence in panel data models. Statistics and Probability Letters 78, 3304–3306. Baltagi, B.H., Song, S.H., Jung, B., Koh, W., 2007. Testing for serial correlation, spatial autocorrelation and random effects using panel data. Journal of Econometrics 140, 5–51. Baltagi, B.H., Song, S.H., Koh, W., 2003. Testing panel data regression models with spatial error correlation. Journal of Econometrics 117, 123–150. Bera, A.K., Yoon, M., 1993. Specification testing with locally misspecified alternatives. Econometric Theory 9, 649–658. Debarsy, N., Ertur, C., 2010. Testing for spatial autocorrelation in a fixed effects panel data model. Regional Science and Urban Economics 40, 453–470. He, M., Lin, K.P., 2012. Testing for spatial autocorrelation in a two-way fixed effects panel data model. Manuscript. He, M., Lin, K.P., 2013. Locally adjusted LM test for spatial dependence in panel data models with fixed effects. Manuscript. Lee, L.F., Yu, J., 2010. Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics 154, 165–185. Qu, X., Lee, L.F., 2012. LM tests for spatial correlation in spatial models with limited dependent variables. Regional Science and Urban Economics 42, 430–445.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Locally adjusted LM test for spatial dependence in fixed effects panel data models Ming He a,∗,1 , Kuan-Pin Lin b,1 a
Department of Economics, Vanderbilt University, VU Station B #351819, 2301 Vanderbilt Place, Nashville, TN 37235-1819, United States
b
Department of Economics, Portland State University, 1721 SW Broadway, Cramer Hall, Suite 241, Portland, OR 97201, United States
highlights • We adopt the spatial panel one-way and two-way fixed effects models. • We construct robust LM tests to guard against local misspecification. • We conduct simulation to show performance of robust LM tests.
article
info
Article history: Received 2 April 2013 Received in revised form 24 June 2013 Accepted 28 June 2013 Available online 11 July 2013 Keywords: Spatial dependence Locally adjusted (robust) LM test Bera and Yoon principle Fixed effects panel data model
abstract Lee and Yu (2010) propose spatial panel data models with one-way and two-way fixed effects. Debarsy and Ertur (2010) construct LM (Lagrange multiplier) and LR (likelihood ratio) tests in the one-way fixed effects model. He and Lin (2012) derive LM tests in the two-way fixed effects model. To guard against possible local misspecification, in this paper we apply Bera and Yoon (1993) principle, and construct locally adjusted (robust) LM tests for spatial dependence in both one-way and two-way fixed effects models. Monte Carlo experiment is carried out to show the advantage of using robust LM tests over the corresponding marginal and conditional versions. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Spatial econometric models have been extensively studied in the past two decades. They are different from the traditional models in the sense that they consider spatial correlation of different observations instead of viewing them as independent. Ignoring spatial effects when they are actually present leads, at best, to inefficient ordinary least squares (OLS) estimators and at worst, to biased and inconsistent OLS estimators. There are two popular forms of spatial correlation. The first is the spatial autoregressive (SAR) process which assumes correlation of dependent variables among neighbors. The second is the spatial error autocorrelation which allows the unobserved error terms to be spatially correlated. Hypothesis testing for spatial dependence has been studied extensively in the literature. For tests of spatial dependence in cross section models, see Anselin (1988a,b, 2001), Anselin et al. (1996),
∗
Corresponding author. Tel.: +1 503 863 9798. E-mail addresses: [email protected] (M. He), [email protected] (K.-P. Lin).
1 The authors would like to thank the anonymous referee for his (her) comments which have greatly helped to improve the paper. 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.06.039
Anselin and Bera (1998) and Qu and Lee (2012). For tests of spatial dependence in panel data models, see Baltagi et al. (2003, 2007, 2013), Baltagi and Liu (2008), Debarsy and Ertur (2010) and He and Lin (2012). Many of the above tests are based on the LM principle. Bera and Yoon (1993) argue that the LM test corresponding to marginal hypothesis might suffer from size distortion under local misspecification of some nuisance parameters, while the LM test corresponding to conditional hypothesis requires maximum likelihood estimation which might be computationally cumbersome. They develop locally adjusted (robust) LM test to guard against local deviation of nuisance parameters from certain pre-specified values. We emphasize two properties of the robust LM test. First, if the null hypothesis is true while the nuisance parameters deviate from prespecified values, the marginal LM test will follow a non-centralized χ 2 distribution asymptotically, tending to reject the null hypothesis too often. However, the robust LM test in this case will follow a centralized χ 2 distribution, thus providing correct asymptotic size, as long as the deviation is local. Second, if the null hypothesis is not true while the nuisance parameters actually take on the pre-specified values, the robust LM test tends to accept the null more often than the marginal LM test, thus it will be less powerful
60
M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
than the marginal LM test. Moreover, robust LM test only requires computing the restricted estimator under relevant joint null hypothesis, and it is the OLS estimator in most cases. Therefore, it is easier to compute than the corresponding conditional LM test. In practice, the robust LM test should be used together with marginal and conditional LM tests in order to make correct statistical inference. In this paper, we extend the works in Debarsy and Ertur (2010) and He and Lin (2012) by deriving robust LM tests for the one-way and two-way fixed effects spatial panel data models developed in Lee and Yu (2010). Since a certain condition (see Section 3.1 below) for the marginal and robust LM tests being equal is not satisfied, then the marginal LM tests constructed in Debarsy and Ertur (2010) and He and Lin (2012) cannot be used to guard against local misspecification. Instead, the robust LM tests should be used. The rest of the paper proceeds as follows: model specifications are briefly reviewed in Section 2. In Section 3, four robust LM tests are provided. In Section 4, we report Monte Carlo experiment results to demonstrate finite sample performance of these robust LM tests. We show the advantage of using robust LM tests by comparing them to the corresponding marginal and conditional LM tests. A conclusion is made in Section 5. 2. Model specification
∗ where Sn (λ) = In −λWn , Rn (ρ) = In −ρ Mn , Vnt = Rn (ρ)[Sn (λ)Ynt∗ − ∗ Xnt β]. √ Similarly, in the two-way fixed effects model, let [Fn,n−1 , ln / n] be the orthonormal eigenvector matrix of the demeaning operator Jn = In − ln l′n /n, where ln is a n × 1 vector of ones. Fn,n−1 is the n × (n − 1) submatrix corresponding to the eigenvalues one. ∗∗ Let Ynt = Fn′ ,n−1 Ynt∗ , Xnt∗∗ = Fn′ ,n−1 Xnt∗ , t = 1, . . . , T . After transformation, the log-likelihood function of the two-way fixed effects model is (n − 1)(T − 1) ln(2π σ 2 ) L2 (θ) = − 2 + (T − 1)[ln |Sn (λ)| + ln |Rn (ρ)|]
− (T − 1)[ln(1 − λ) + ln(1 − ρ)] −
T −1 1 ∗ ∗′ ∗∗ Vnt Vnt 2
2σ
(4)
t =1
∗∗ where Vnt = R∗n (ρ)[Sn∗ (λ)Ynt∗∗ − Xnt∗∗ β], R∗n (ρ) = In−1 − ρ Mn∗ , Sn∗ (λ) = In−1 − λWn∗ , Mn∗ = Fn′ ,n−1 Mn Fn,n−1 , Wn∗ = Fn′ ,n−1 Wn Fn,n−1 . Lee and Yu (2010) show that the maximum likelihood estimators based on the log-likelihood functions (3) and (4), under a set of regularity assumptions, are consistent and centered at the true values of population parameters.
In this paper, we consider fixed effects panel data models with first order SAR lag and first order SAR disturbances. Using the notations in Lee and Yu (2010), the SAR panel data model with individual effects (one-way fixed effects) and SAR disturbances is
3. Bera and Yoon principle, robust LM tests
Ynt = λ0 Wn Ynt + Xnt β0 + cn0 + Unt
Bera and Yoon (1993) discuss the issue of local misspecification regarding to some nuisance parameters. Let θ = (γ ′ , ψ ′ , φ ′ )′ and the log-likelihood function be L(θ), and denote γ as the restricted estimator of γ under the joint null hypothesis H0 : ψ = ψ0 , φ = φ0 . Let θ = ( γ ′ , ψ0′ , φ0′ )′ and Iψγ ′ = −E [∂ 2 L(θ)/∂ψ∂γ ′ ], etc. The robust LM test statistic corresponding to the marginal null hypothesis H0 : ψ = ψ0 (φ = φ0 ) is given by
Unt = ρ0 Mn Unt + Vnt
(1)
t = 1, 2, . . . , T
while the SAR panel data model with both individual and time effects (two-way fixed effects) and SAR disturbances is Ynt = λ0 Wn Ynt + Xnt β0 + cn0 + αt0 ln + Unt Unt = ρ0 Mn Unt + Vnt
(2)
t = 1, 2, . . . , T
where Ynt = (y1t , y2t , . . . , ynt ) and Vnt = (v1t , v2t , . . . , vnt ) are n × 1 vectors for period t, vit is i.i.d. across i and t with zero mean and variance σ02 . Xnt is a n × k matrix of time varying explanatory variables for period t. cn0 is a n × 1 vector representing the individual fixed effects, while αt0 represents the time effects in period t. Wn , Mn are exogenous spatial weights matrices associated with spatially correlated dependent variable and spatially correlated error. Lee and Yu (2010) show that the maximum likelihood estimator (MLE) based on a direct approach (i.e. MLE from the above two models) does not consistently estimate σ02 unless T is large. To solve this problem, they propose a certain transformation approach to eliminate the√ fixed individual and time effects. Specifically, let [FT ,T −1 , lT / T ] be the orthonormal eigenvector matrix of the demeaning operator JT = IT − lT l′T /T , where IT is an identity matrix of size T , and lT is a T × 1 vector of ones. FT ,T −1 is the T × (T − 1) submatrix corresponding to the eigenvalues ′ ′ ′ ′ ∗′ , . . . , YnT ) , where ⊗ one. Let (Yn1 , . . . , Yn∗,T −1 )′ = (FT′ ,T −1 ⊗ In )(Yn1 denotes the Kronecker product, and In is an identity matrix of size ∗ n. Xnt , t = 1, . . . , T are defined similarly (see Lee and Yu (2010) for more details of the transformation approach and the regularity assumptions). Let θ = (β′ , λ, ρ, σ 2 )′ , then after transformation, the loglikelihood function of the one-way fixed effects model is ′
L1 (θ) = −
n(T − 1) 2
′
T −1 1 ∗′ ∗ Vnt Vnt 2
2σ
t =1
LM∗ψ =
∂ L∗ ∂ψ
′
(I∗ )−1
∂ L∗ ∂ψ
θ
−1 ∗ −1 where (∂ L∗ /∂ψ)|θ = (∂ L/∂ψ − Iψφ ′ ·γ Iφφ ′ ·γ ∂ L/∂φ)| θ , (I ) | θ =
−1 −1 −1 (Iψψ ′ ·γ − Iψφ ′ ·γ Iφφ |θ and Iψψ ′ ·γ = Iψψ ′ − Iψγ ′ Iγγ ′ ·γ Iφψ ′ ·γ ) ′ −1 ′ ′ ′ ′ ′ ′ ′ ′ Iγψ , Iφφ ·γ = Iφφ − Iφγ Iγγ ′ Iγφ , Iψφ ·γ = Iφψ ′ ·γ = Iψφ −
−1 Iψγ ′ Iγγ ′ Iγφ ′ . The crucial quantity in the above formula is Iψφ ′ ·γ . If Iψφ ′ ·γ = 0, then LM∗ψ = LMψ , where LMψ denotes the marginal
LM test for ψ . Bera and Yoon (1993) also show the identity that LMψφ = LMφ + LM∗ψ = LMψ + LM∗φ , where LMψφ , LMφ , LM∗φ denote the joint LM test for ψ, φ , the marginal and robust LM tests for φ , respectively. Thus if Iψφ ′ ·γ = 0, LMψφ = LMφ + LMψ . The advantage of the robust LM test is that it only requires estimation under the joint null hypothesis, thus is simpler, and it can provide valid asymptotic size when the nuisance parameters deviate locally from the pre-specified values (for example, φ deviates locally from φ0 in the case of marginal test for ψ ). However, when indeed φ = φ0 in the population model, LM∗ψ will generally be less powerful than LMψ . 3.2. Robust LM tests in the one-way fixed effects model In this section, we consider two robust LM tests in the one-way fixed effects model, two corresponding hypotheses are considered. Hypothesis (1.1): H01.1 : λ = 0 (allowing for local deviation of ρ from 0) vs. H11.1 : otherwise.
ln(2π σ 2 ) + (T − 1)[ln |Sn (λ)|
+ ln |Rn (ρ)|] −
3.1. Bera and Yoon principle
(3)
Hypothesis (1.2): H01.2 : ρ = 0 (allowing for local deviation of λ from 0) vs. H11.2 : otherwise.
M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
The corresponding joint null hypothesis is H0 : ρ = λ = 0. The robust LM tests are given by3
z1λ − LM∗1λ = ξ1λ LM∗1ρ
= ξ1ρ z1ρ −
b12 b13
2 z1ρ
, 2
b12 b11 + ω1
z1λ
where ξ1λ = b13 /(b11 b13 − b212 + b13 ω1 ), ξ1ρ = (b11 + ω1 )/
∗′ ∗ (b11 b13 − b212 + b13 ω1 ), z1λ = (1/ σ12 ) Tt =−11 Vnt Wn Ynt , z1ρ = (1/ σ12 ) T −1 ∗′ ∗ ′ t =1 Vnt Mn Vnt , b11 = (T − 1)tr[(Wn + Wn )Wn ], b12 = (T − 1)tr[(Mn + Mn′ )Wn ], b13 = (T − 1)tr[(Mn + Mn′ )Mn ], ω = (1/ σ 2) T −11 ∗′ ∗ 1−1 T −1 ∗′ T −1 2 ∗ ′ ∗ ∗ ′ Ynt ) ( t =1 Xnt Xnt ) Y )−(1/ σ1 )( t =1 Xnt Wn Y ) (W (W t =1 n′ nt ∗ n nt ∗ ∗ ( tT=−11 Xnt∗ Wn Ynt ), and Vnt = Ynt∗ − Xnt∗ β1 , Ynt = Xnt∗ β1 , β1 = T − 1 ∗′ ∗ − 1 T − 1 ∗′ ∗ ′ T − 1 2 ∗ ∗ ( t =1 Xnt Xnt ) ( t =1 Xnt Ynt ), σ1 = t =1 Vnt Vnt /[n(T − 1)]. ′ ′ Let θ 1 = ( β 1 , 0, 0, σ12 )′ , then γ 1 = ( β1 , σ12 )′ . In hypothesis (1.1), ψ = λ, φ = ρ , and in hypothesis (1.2), ψ = ρ, φ = λ. It can be shown that (I1λρ·γ )|θ 1 = (I1ρλ·γ )|θ 1 = b12 ̸= 0, then ∂ L1 / ∂λ and ∂ L1 /∂ρ are partially correlated even after eliminating the effects of ∂ L1 /∂γ on ∂ L1 /∂λ and ∂ L1 /∂ρ , rendering the differences between the robust LM tests and the marginal LM tests, which are given by LM1λ =
1 b11 + ω1
z12λ ,
LM1ρ =
1 b13
z12ρ .
On one hand, if the data is generated according to λ = 0 (ρ ̸= 0), the asymptotic size of LM1λ will be distorted, whereas LM∗1λ converges to a centralized χ12 distribution under local deviation of ρ from 0. In this situation, LM∗1λ should be used for statistical inference. On the other hand, if ρ = 0 in the population model, LM∗1λ will generally be less powerful than LM1λ . A similar discussion applies to LM∗1ρ and LM1ρ .
between the robust LM tests and the marginal LM tests, which are given by LM2λ =
1 b21 + ω2 ∗
In this section, we consider two robust LM tests in the two-way fixed effects model, two corresponding hypotheses are considered. Hypothesis (2.1): H02.1 : λ = 0 (allowing for local deviation of ρ from 0) vs. H12.1 : otherwise. Hypothesis (2.2): H02.2 : ρ = 0 (allowing for local deviation of λ from 0) vs. H12.2 : otherwise. The corresponding joint null hypothesis is H0 : ρ = λ = 0. The robust LM tests are given by b∗ z2ρ LM∗2λ = ξ2λ z2λ − ∗22 b23
2
,
b∗ LM∗2ρ = ξ2ρ z2ρ − ∗ 22 z2λ ω2 b21 +
2
ω2 )/ ω2 ), ξ2ρ = (b∗21 + where ξ2λ = b∗23 /(b∗21 b∗23 − b∗222 + b∗23
∗ ∗′ Wn∗ ω2 ), z2λ = (T − 1) + (1/ σ22 ) Tt =−11 Vnt (b∗21 b∗23 − b∗222 + b∗23 ′ T −1 ∗ ∗ 2 ∗∗∗ ∗ ∗∗ Ynt , z2ρ = (T − 1) + (1/ σ2 ) t =1 Vnt Mn Vnt , b2j = b2j − 2(T − ′ 1)/(n − 1), j = 1, 2, 3, b21 = (T − 1)tr[(Wn∗ + Wn∗ )Wn∗ ], b22 = ′ ′ (T − 1)tr[(Mn∗ + Mn∗ )Wn∗ ], b23 = (T − 1)tr[(Mn∗ + Mn∗ )Mn∗ ], and V ∗∗ T −1 ∗ ∗∗ ′ nt∗ 2 ∗∗ ∗∗ ∗∗ ∗∗ = Ynt − Xnt β2 , Ynt = Xnt β2 , ω2 = (1/ σ2 ) t =1 (Wn Ynt ) (Wn ′ ′ ′ ∗∗ ∗∗ ′ Ynt ) − (1/ σ22 )( Tt =−11 Xnt∗ ∗ Wn∗ Ynt ) ( Tt =−11 Xnt∗ ∗ Xnt∗∗ )( Tt =−11 Xnt∗ ∗ Wn∗
3 We do not present the derivation of test statistics in this paper, but it can be found in the earlier manuscript He and Lin (2013).
z22λ ,
LM2ρ =
1 b∗23
z22ρ .
On one hand, if the data is generated according to λ = 0 (ρ ̸= 0), the asymptotic size of LM2λ will be distorted, whereas LM∗2λ converges to a centralized χ12 distribution under local deviation of ρ from 0. In this situation, LM∗2λ should be used for statistical inference. On the other hand, if ρ = 0 in the population model, LM∗2λ will generally be less powerful than LM2λ . A similar discussion applies to LM∗2ρ and LM2ρ . 4. Monte Carlo simulations In this section, we conduct a small Monte Carlo simulation to evaluate the size and power performance of the proposed test statistics. The data generating process is, for t = 1, . . . , T , Ynt = λ0 Wn Ynt + Xnt1 β10 + Xnt2 β20
+ cn0 + αt0 ln + (In − ρ0 Mn )−1 Vnt the time effects term—αt0 is suppressed in the one-way fixed effects model. Vnt = (v1t , v2t , . . . , vnt )′ , vit ’s are i.i.d. across i and t. We generate vit from a mixed distribution, with probability 0.5 from a log-normal distribution, and with probability 0.5 from a Weibull distribution. Specifically,
vit = rit − E [rit ], 3.3. Robust LM tests in the two-way fixed effects model
61
′ ′ ∗∗ ∗ ∗′ Ynt ), β2 = ( tT=−11 Xnt∗ ∗ Xnt∗∗ )−1 ( Tt =−11 Xnt∗ ∗ Ynt∗∗ ), σ22 = Tt =−11 Vnt ∗∗ Vnt /[(n − 1)(T − 1)]. ′ ′ Let θ 2 = ( β 2 , 0, 0, σ22 )′ , then γ 2 = ( β2 , σ22 )′ . In hypothesis (2.1), ψ = λ, φ = ρ , and in hypothesis (2.2), ψ = ρ, φ = λ. It can be shown that (I2λρ·γ )|θ 2 = (I2ρλ·γ )|θ 2 = b∗22 ̸= 0, then ∂ L2 /∂λ and ∂ L2 /∂ρ are partially correlated even after eliminating the effects of ∂ L2 /∂γ on ∂ L2 /∂λ and ∂ L2 /∂ρ , rendering the differences
rit = (1 − Wit )Z1,it + Wit Z2,it ,
where Wit is a Bernoulli random variable with success probability p = 0.5, log Z1,it ∼ N (−2, 4), Z2,it is generated from a Weibull distribution with scale parameter 5 and shape parameter 1, Wit , Z1,it , Z2,it are independent of each other. rit is demeaned by the population mean so that E [vit ] = 0. For the individual effects, each element of cn0 is generated independently from a uniform distribution on [−5, 5]. For the time effects, α(t +1)0 = 1.05αt0 , α10 is generated from a uniform distribution on [0, 10]. Each element of Xnt1 is generated independently from N (0, 16), while each element of Xnt2 is generated independently from a uniform distribution on [0, 10]. For the spatial weights matrices, Wn = Mn , and they both take the form of a first order rook contiguity matrix. For the parameter values, we set β10 = 0.5, β20 = 0.7, both λ0 and ρ0 take values in [−0.8, 0.8], with increment 0.2. For each combination of parameter values, two sample sizes are chosen, namely, n = 81, T = 8, and n = 196, T = 10, with 1000 replications performed. The nominal size is set to be 0.05. The empirical size and power of the marginal and robust LM tests in the one-way fixed effects model is summarized in Table 1. We compare the empirical size of the marginal and robust LM tests under local misspecification, and their empirical power without local misspecification. For the robust test LM∗1λ , when λ = 0 (ρ ̸= 0), it follows a centralized χ 2 distribution asymptotically. It shows strength in correctly accepting the null hypothesis λ = 0, while LM1λ incorrectly reject the null hypothesis. For example, when the sample size is (81, 8) and λ = 0 (ρ = 0.4), the frequency of rejection (FR) of LM∗1λ is 0.065, while the FR of LM1λ is 1. Generally, the empirical size of LM∗1λ is reasonable when |ρ| ≤ 0.4, this is because we only allow for local deviation of ρ from 0. The size performance of LM∗1λ is similar for the (196, 10) sample. Therefore, when
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M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
Table 1 Size and power comparison of marginal and robust LM tests, one-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ=0 ρ
Size LM1λ
LM∗1λ
LM1ρ
LM∗1ρ
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.917 0.029 0.925 1.000 1.000 1.000
0.195 0.111 0.068 0.064 0.036 0.065 0.065 0.101 0.136
1.000 1.000 1.000 0.955 0.033 0.963 1.000 1.000 1.000
0.952 0.918 0.697 0.264 0.039 0.252 0.733 0.913 0.951
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 1.000 0.046 1.000 1.000 1.000 1.000
0.161 0.117 0.071 0.051 0.049 0.053 0.075 0.088 0.154
1.000 1.000 1.000 1.000 0.041 1.000 1.000 1.000 1.000
0.992 0.989 0.956 0.595 0.042 0.569 0.953 0.989 0.993
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
Power
ρ=0 λ
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Size
Power
Table 2 Size and power comparison of robust and conditional LM tests, one-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ
LM1ρ
LM∗1ρ
LM1λ
LM∗1λ
1.000 1.000 1.000 0.941 0.033 0.968 1.000 1.000 1.000
0.035 0.050 0.042 0.045 0.039 0.053 0.049 0.041 0.030
1.000 1.000 1.000 0.968 0.029 0.983 1.000 1.000 1.000
0.979 0.963 0.806 0.322 0.036 0.325 0.802 0.946 0.976
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.999 0.041 1.000 1.000 1.000 1.000
0.046 0.057 0.059 0.055 0.042 0.041 0.055 0.052 0.035
1.000 1.000 1.000 0.999 0.046 1.000 1.000 1.000 1.000
0.994 0.991 0.965 0.624 0.049 0.631 0.961 0.993 0.996
−0.8 −0.6 −0.4 −0.2
λ = 0 (ρ ̸= 0), using marginal test LM1λ will be misleading. However, there is one drawback of applying the robust LM test, that is, when the assumption of the nuisance parameter (ρ = 0 in this case) is indeed true, LM∗1λ is less powerful than the marginal test LM1λ . For instance, when λ = −0.4 (ρ = 0) and the sample size is (81, 8), the FR of LM1λ is 1, while that of LM∗1λ is 0.806. However, the loss of power is not very severe, and it improves as n becomes larger. For example, when λ = −0.4 (ρ = 0), the FR of LM∗1λ is 0.965 for the (196, 10) sample. The performance of LM∗1ρ follows a similar discussion as that of LM∗1λ , and its size performance is even better in the sense that it provides correct asymptotic size even when λ deviates far away from 0. In the one-way fixed effects model, we also compare performances of the robust LM tests with those of the corresponding conditional LM tests. The results are summarized in Table 2. For the sake of compactness, we only report the results when the nuisance parameters take values −0.4, 0.4. LM1λ|ρ and LM1ρ|λ are the conditional LM tests in Debarsy and Ertur (2010). In the left four columns, generally, the size and power performances of LM∗1λ and LM1λ|ρ are comparable. For example, when the sample size is (81, 8) and ρ = 0.4, the empirical size of LM∗1λ is 0.065, while that of LM1λ|ρ is 0.060. Both LM∗1λ and LM1λ|ρ are a little oversized for the (196, 10) sample. For instance, when ρ = −0.4, the empirical size of LM∗1λ is 0.071, and that of LM1λ|ρ is 0.080. There is an interesting pattern in their power performances. When ρ = −0.4, the power of LM∗1λ is lower than that of LM1λ|ρ when λ < 0, while the reverse order holds when λ > 0. And there is a similar but reverse pattern when ρ = 0.4. Also, the FR of LM1λ|ρ becomes very low when λ = 0.8 (ρ = −0.4) and λ = −0.8 (ρ = 0.4), although this loss in power improves for the (196, 10) sample. A similar discussion applies to the comparison between LM∗1ρ and LM1ρ|λ . The performances and comparison of tests in the two-way fixed effects model are similar to those in the one-way fixed effects model, and they are summarized in Tables 3 and 4. In general, the robust tests LM∗1λ , LM∗1ρ , LM∗2λ , LM∗2ρ have very good performances in correctly accepting the null hypothesis when the nuisance parameters deviate from pre-specified values. Although there is a loss in power compared to the marginal LM tests when the assumption of the nuisance parameters is true, the loss is not very severe, and it improves as the sample size becomes larger. Also, when the nuisance parameters deviate from pre-specified values, performances of the robust LM tests are generally comparable to those of the conditional LM tests. Finally, the robust LM tests are computationally easier to obtain than the
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
ρ = −0.4
ρ = 0.4
LM∗1λ
LM1λ|ρ
LM∗1λ
LM1λ|ρ
ρ
0.938 0.848 0.684 0.302 0.068 0.318 0.805 0.970 0.984
0.982 0.974 0.797 0.355 0.059 0.356 0.793 0.650 0.248
0.993 0.971 0.820 0.307 0.065 0.284 0.662 0.897 0.933
0.362 0.654 0.796 0.391 0.060 0.371 0.830 0.971 0.978
−0.8 −0.6 −0.4 −0.2
0.992 0.972 0.909 0.549 0.071 0.582 0.958 0.985 0.997
1.000 0.996 0.971 0.731 0.080 0.746 0.968 0.873 0.495
0.998 0.987 0.971 0.589 0.075 0.530 0.917 0.980 0.989
0.608 0.829 0.965 0.751 0.068 0.689 0.973 0.997 1.000
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
λ = −0.4
λ = 0.4
LM∗1ρ
LM1ρ|λ
LM∗1ρ
LM1ρ|λ
0.747 0.759 0.540 0.199 0.042 0.264 0.769 0.951 0.981
0.991 0.966 0.792 0.319 0.061 0.373 0.803 0.806 0.261
0.976 0.955 0.745 0.236 0.049 0.205 0.546 0.763 0.760
0.306 0.812 0.792 0.379 0.049 0.319 0.779 0.953 0.993
0.956 0.971 0.900 0.467 0.059 0.542 0.966 0.992 0.997
1.000 0.997 0.972 0.688 0.077 0.720 0.960 0.926 0.400
0.991 0.992 0.957 0.523 0.055 0.508 0.909 0.973 0.974
0.386 0.913 0.968 0.714 0.085 0.677 0.972 0.997 1.000
Table 3 Size and power comparison of marginal and robust LM Tests, two-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ=0 ρ
Size LM2λ
LM∗2λ
LM2ρ
LM∗2ρ
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 0.916 0.034 0.899 1.000 1.000 1.000
0.194 0.112 0.065 0.059 0.034 0.059 0.063 0.088 0.121
1.000 1.000 1.000 0.952 0.036 0.946 1.000 1.000 1.000
0.951 0.907 0.685 0.267 0.039 0.226 0.711 0.908 0.953
−0.8 −0.6 −0.4 −0.2
1.000 1.000 1.000 1.000 0.046 1.000 1.000 1.000 1.000
0.165 0.119 0.074 0.049 0.048 0.059 0.070 0.096 0.144
1.000 1.000 1.000 1.000 0.043 1.000 1.000 1.000 1.000
0.992 0.989 0.953 0.595 0.050 0.567 0.956 0.989 0.994
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
Power
ρ=0 λ
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Size
Power
LM2ρ
LM∗2ρ
LM2λ
LM∗2λ
1.000 1.000 1.000 0.934 0.036 0.965 1.000 1.000 1.000
0.037 0.046 0.043 0.048 0.039 0.044 0.046 0.039 0.027
1.000 1.000 1.000 0.965 0.034 0.981 1.000 1.000 1.000
0.978 0.960 0.805 0.316 0.034 0.318 0.780 0.939 0.971
1.000 1.000 1.000 0.999 0.043 1.000 1.000 1.000 1.000
0.047 0.055 0.060 0.053 0.050 0.039 0.045 0.050 0.038
1.000 1.000 1.000 0.999 0.046 1.000 1.000 1.000 1.000
0.994 0.989 0.965 0.622 0.048 0.626 0.962 0.993 0.994
Table 4 Size and power comparison of robust and conditional LM tests, two-way effects model, sample sizes: upper part: (81, 8), lower part: (196, 10).
λ −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8
ρ = −0.4
ρ = 0.4
LM∗2λ
LM2λ|ρ
LM∗2λ
LM2λ|ρ
ρ
0.937 0.842 0.668 0.301 0.065 0.309 0.781 0.965 0.982
0.982 0.972 0.779 0.333 0.063 0.343 0.764 0.667 0.187
0.992 0.971 0.818 0.307 0.063 0.273 0.649 0.894 0.934
0.320 0.648 0.785 0.368 0.065 0.351 0.819 0.962 0.954
−0.8 −0.6 −0.4 −0.2
0.992 0.971 0.907 0.533 0.074 0.575 0.955 0.985 0.997
1.000 0.997 0.972 0.711 0.079 0.723 0.966 0.877 0.416
0.998 0.986 0.970 0.583 0.070 0.511 0.901 0.977 0.988
0.581 0.829 0.963 0.746 0.071 0.687 0.969 0.997 1.000
−0.8 −0.6 −0.4 −0.2
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
λ = −0.4
λ = 0.4
LM∗2ρ
LM2ρ|λ
LM∗2ρ
LM2ρ|λ
0.736 0.758 0.546 0.198 0.043 0.249 0.754 0.945 0.980
0.992 0.964 0.772 0.316 0.064 0.358 0.792 0.812 0.347
0.972 0.944 0.728 0.235 0.046 0.207 0.533 0.760 0.801
0.285 0.799 0.765 0.379 0.055 0.306 0.770 0.940 0.993
0.956 0.970 0.899 0.461 0.060 0.536 0.963 0.991 0.997
1.000 0.996 0.973 0.675 0.073 0.716 0.965 0.931 0.441
0.991 0.994 0.951 0.515 0.045 0.478 0.908 0.973 0.976
0.379 0.909 0.966 0.702 0.085 0.675 0.969 0.997 1.000
M. He, K.-P. Lin / Economics Letters 121 (2013) 59–63
conditional LM tests since they only require OLS estimation, which greatly saves the computation time. For instance, when the sample size is (196, 10), the simulation time of LM∗1λ is 826 s in the oneway effects model, while that of LM1λ|ρ is 14 859 s, which is almost 18 times larger than that of LM∗1λ . In sum, researchers are recommended to use the robust LM tests together with the marginal and conditional LM tests to draw correct statistical inference. 5. Conclusion In this paper, we adopt the spatial panel data models with oneway and two-way fixed effects in Lee and Yu (2010). We construct the robust LM test statistics based on the Bera and Yoon (1993) principle. To complement the works in Debarsy and Ertur (2010) and He and Lin (2012), we provide the robust LM tests for a spatially correlated dependent variable allowing for the local presence of spatially correlated error and the robust LM tests for spatially correlated error allowing for the local presence of a spatially correlated dependent variable. Monte Carlo simulations show the advantage of using robust LM tests over the corresponding marginal and conditional LM tests. In practice, researchers are suggested to use robust LM tests together with marginal and conditional LM tests to draw correct statistical inference. Finally, in our Monte Carlo study, we find that the performance of the robust LM tests for a spatially correlated error is generally better than that of the robust LM tests for a spatially correlated dependent variable.
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