Testing for Random Effects and Spatial Lag Dependence in Panel Data Models

Syracuse University

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Economics - Faculty Scholarship

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_Affairs

3-1-2008

Testing for Random Effects and Spatial Lag Dependence in Panel

Testing for Random Effects and Spatial Lag Dependence in Panel

Data Models

Data Models

Badi Baltagi

Long Liu

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Recommended Citation

Recommended Citation

Baltagi, Badi and Liu, Long, "Testing for Random Effects and Spatial Lag Dependence in Panel Data Models" (2008). Economics - Faculty Scholarship. 90.

https://surface.syr.edu/ecn/90

Electronic copy available at: http://ssrn.com/abstract=1809089

ISSN: 1525-3066

Center for Policy Research

Working Paper No. 102

TESTING FOR RANDOM EFFECTS AND

SPATIAL LAG DEPENDENCE IN PANEL

DATA MODELS

Badi H. Baltagi and Long Liu

Center for Policy Research

Maxwell School of Citizenship and Public Affairs

Syracuse University

426 Eggers Hall

Syracuse, New York 13244-1020

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e-mail: [email protected]

March 2008

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Professor of Economics & Public Administration

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Professor of Public Administration Professor of Economics and Public Administration Associate Director, Aging Studies Program Associate Director, Metropolitan Studies Program

SENIOR RESEARCH ASSOCIATES

Badi Baltagi ... Economics Kalena Cortes………Education Thomas Dennison ... Public Administration William Duncombe ... Public Administration Gary Engelhardt ... Economics Deborah Freund ... Public Administration Madonna Harrington Meyer ... Sociology Christine Himes ... Sociology William C. Horrace ... Economics Duke Kao ... Economics Eric Kingson ... Social Work Thomas Kniesner ... Economics Jeffrey Kubik ... Economics Andrew London ... Sociology

Len Lopoo ... Public Administration Amy Lutz……… Sociology Jerry Miner ... Economics Jan Ondrich ... Economics John Palmer ... Public Administration Lori Ploutz-Snyder ... Exercise Science David Popp ... Public Administration Christopher Rohlfs ... Economics Stuart Rosenthal ... Economics Ross Rubenstein ... Public Administration Perry Singleton………Economics Margaret Usdansky ... Sociology Michael Wasylenko ... Economics Janet Wilmoth ... Sociology

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Amy Agulay……….Public Administration Javier Baez ... Economics Sonali Ballal ... Public Administration Jesse Bricker ... Economics Maria Brown ... Social Science Il Hwan Chung………Public Administration Mike Eriksen ... Economics Qu Feng ... Economics Katie Fitzpatrick... Economics Chantell Frazier………...Sociology Alexandre Genest ... Public Administration Julie Anna Golebiewski...Economics

Nadia Greenhalgh-Stanley ... Economics Sung Hyo Hong ... Economics Neelakshi Medhi ... Social Science Larry Miller ... Public Administration Phuong Nguyen ... Public Administration Wendy Parker ... Sociology Shawn Rohlin ... Economics Carrie Roseamelia ... Sociology Jeff Thompson ... Economics Coady Wing ... Public Administration Ryan Yeung ... Public Administration Can Zhou ... Economics

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Abstract

This paper derives a joint Lagrange Multiplier (LM) test which simultaneously tests for

the absence of spatial lag dependence and random individual effects in a panel data regression

model. It turns out that this LM statistic is the sum of two standard LM statistics. The first one

tests for the absence of spatial lag dependence ignoring the random individual effects, and the

second one tests for the absence of random individual effects ignoring the spatial lag

dependence. This paper also derives two conditional LM tests. The first one tests for the absence

of random individual effects without ignoring the possible presence of spatial lag dependence.

The second one tests for the absence of spatial lag dependence without ignoring the possible

presence of random individual effects.

JEL codes: C12 and C23

Testing For Random E¤ects and Spatial Lag Dependence in Panel

Data Models

Badi H. Baltagi, Long Liu

Syracuse University

March 25, 2008

Abstract

This paper derives a joint Lagrange Multiplier (LM) test which simultaneously tests for the absence of spatial lag dependence and random individual e¤ects in a panel data regression model. It turns out that this LM statistic is the sum of two standard LM statistics. The …rst one tests for the absence of spatial lag dependence ignoring the random individual e¤ects, and the second one tests for the absence of random individual e¤ects ignoring the spatial lag dependence. This paper also derives two conditional LM tests. The …rst one tests for the absence of random individual e¤ects without ignoring the possible presence of spatial lag dependence. The second one tests for the absence of spatial lag dependence without ignoring the possible presence of random individual e¤ects.

Key Words: Panel Data; Spatial Lag Dependence; Lagrange Multiplier Tests; Random E¤ ects.

1

Introduction

Spatial models deal with correlation across spatial units usually in a cross-section setting, see Anselin (1988a). Panel data models allow the researcher to control for heterogeneity across these units, see Baltagi (2005). Spatial panel models can control for both heterogeneity and spatial correlation, see Baltagi, Song and Koh (2003). Testing for spatial dependence has been extensively studied by Anselin (1988a, 1988b, 2001) and Anselin and Bera (1998), to mention a few. Baltagi, Song and Koh (2003) considered the problem of jointly testing for random region e¤ects in the panel as well as spatial correlation across these regions. However, the last study allowed for spatial correlation only in the remainder error term. This paper generalizes the Baltagi, Song and Koh (2003) to allow for spatial lag dependence of the autoregressive kind in the dependent variable rather than the error term. In fact, this paper derives a joint LM test which simultaneously tests for the absence of spatial lag dependence and random individual e¤ects in a panel data regression model. It

Address correspondence to: Badi H. Baltagi, Center for Policy Research, 426 Eggers Hall, Syracuse University, Syracuse, NY 13244-1020; e-mail: [email protected].

y_{Long Liu: Economics Department, 110 Eggers Hall, Syracuse University, Syracuse, NY 13244-1020; e-mail:}

turns out that this LM statistic is the sum of two standard LM statistics. The …rst LM, tests for the absence of spatial lag dependence ignoring the random individual e¤ects. This is the standard LM test derived in Anselin (1988b) for cross-section data. The second LM, tests for the absence of random individual e¤ects ignoring the spatial lag dependence. This is the standard LM test derived in Breusch and Pagan (1980) for panel data. This paper also derives two conditional LM tests. The …rst one tests for the absence of random individual e¤ects without ignoring the possible presence of spatial lag dependence. The second one tests for the absence of spatial lag dependence without ignoring the possible presence of random individual e¤ects. This should provide useful diagnostics for applied researchers working in this area.

2

The model and test statistics

Consider a panel data regression model with spatial lag dependence:

yt= W yt+ Xt + ut; i = 1; : : : ; N ; t = 1; :::; T (1)

where yt0 = (yt1; : : : ; ytN) is a vector of observations on the dependent variables for N regions or households

at time t = 1; :::; T: is a scalar spatial autoregressive coe¢ cient and W is a known N N spatial weight matrix whose diagonal elements are zero. W also satis…es the condition that (IN W ) is non-singular for

all j j < 1: IN is an identity matrix of dimension N . Xtis an N k matrix of observations on k explanatory

variables at time t. u0

t= (ut1; : : : ; utN) is a vector of disturbances following an error component model:

ut= + t (2)

where 0 _{= (}

1; : : : ; N) and i is i.i.d. over i and is assumed to be N (0; 2): 0t= ( t1; : : : ; tN) and ti is

i.i.d. over t and i and is assumed to be N (0; 2_{). The f}

ig process is also independent of the f itg process.

Equation (1) can be rewritten in matrix notation as

y = (IT W ) y + X + u; i = 1; : : : ; N ; t = 1; :::; T (3)

where y is of dimension N T 1, X is N T k, is k 1 and u is N T 1. The observations are ordered with t being the slow running index and i the fast running index, i.e., y0 _{= (y}

11; : : : ; y1N; : : : ; yT 1; : : : ; yT N) : X is

assumed to be of full column rank and its elements are assumed to be asymptotically bounded in absolute value. Equation (2) can also be written in vector form as

u = ( T IN) + ; (4)

where 0 = ( 0₁; : : : ; 0_T) ; T is a vector of ones of dimension T , IN is an identity matrix of dimension N;

and denotes the Kronecker product. Under these assumptions, the variance-covariance matrix for u can be written as

= 2(JT IN) + 2(IT IN) ; (5)

where JT is a matrix of ones of dimension T .

Under the normality assumption, the log-likelihood function of equation (1) is given by

L = N T 2 ln 2 1 2ln j j + T ln jAj 1 2[(IT A) y X ] 0 1_[(I T A) y X ] (6)

where A = IN W: Ord (1975) shows that ln jIN W j =PNi=1ln (1 !i), where !i’s are the eigenvalues

of W . Using the notation in Baltagi (2005), we can write = 2 ; where = Q + 2P; P = JT IN; JT = T 0T=T; Q = IT N P; 2= 2= 21and 21= T 2+ 2. From which it follows that ln j j = NT ln 2+N ln 2.

The log-likelihood function in (6) can be rewritten as

L = N T 2 ln 2 1 2 N T ln 2_{+ N ln} 2 _+T N X i=1 ln (1 !i) 1 2 2[(IT A) y X ] 0 1_[(I T A) y X ] (7) and one can estimate this model using maximum likelihood, see Anselin (1988a).

This paper derives a joint LM test for the absence of spatial lag dependence as well as random e¤ects. The null hypothesis is H0a : = 2 = 0; and the alternative H1a is that at least one component is not zero.

This generalizes the LM test derived in Anselin (1988b) for the absence of spatial lag dependence Hb 0 : = 0

(assuming no random e¤ects, i.e., 2 _{= 0), and the Breusch and Pagan (1980) LM test for the absence of}

random e¤ects Hc

0 : 2 = 0 (assuming no spatial lag dependence, i.e., = 0). We also derive two conditional

LM tests, one for Hd

0 : = 0 (assuming the possible existence of random e¤ects, i.e., 2 > 0); and the other

one for He

0 : 2 = 0 (assuming the possible existence of spatial lag dependence, i.e., may be di¤erent from

zero). All the proofs are given in the Appendix to the paper.

2.1

Joint LM test for

H

0

:

=

= 0

The joint LM test statistic for testing Ha

0 : = 2 = 0 is given by LMJ= R2 B + N T 2 (T 1)G 2_{= LM + LM (8)}

where B = T tr W2+ W0W +_e 2e0X0(IT W0) M (IT W ) X e; M = I X (X0X) 1X0; G = Tu~ 0_{P ~u} ~ u0_u~ 1; R = N Tu~0(IT W )y ~

u0_u~ : LM = R2=B; and LM = N T G2=2 (T 1) : e is the restricted MLE under H0a which

yields OLS, ~u denotes the OLS residuals, and _e2= ~u0_{u=N T . R is a generalization of a similar term de…ned~}

in Anselin (1988b) for the LM test of no spatial dependence in the cross-section case. In fact, R can be interpreted as N T times the regression coe¢ cient of (IT W ) y on ~u: Here, the joint LM test LMJ is

the sum of two LM test statistics: The …rst is LM = R2_{=B;which is the LM test statistic for testing}

Hb

0 : = 0 assuming there is no random region e¤ects, i.e., assuming 2 = 0, see Anselin (1988a). LM is

asymptotically distributed as 2

1 under H0b: The second is LM = 2(TN T1)G2;which is the LM test statistic

for testing Hc

0 : 2 = 0 assuming there is no spatial lag dependence, i.e., assuming that = 0, see Breusch

and Pagan (1980). Since LM and LM are asymptotically independent, LMJ is asymptotically distributed

as 22 under H0a. It is important to point out that the asymptotic distribution of our test statistics are not

explicitly derived in the paper but that they are likely to hold under a similar set of primitive assumptions developed by Kelejian and Prucha (2001).

2.2

Conditional LM Test for

H

:

= 0 (assuming

> 0)

When one uses LM de…ned in (8) to test Hb

0: = 0, one implicitly assumes that the random region e¤ects

do not exist. This may lead to incorrect inference especially when 2 _{is large. To overcome this problem, we}

derive a conditional LM test for no spatial lag dependence assuming the possible existence of random region e¤ects. The null hypothesis is Hd

0 : = 0 (assuming 2 > 0), and the conditional LM test statistic is given

by LM= = R21=B1; (9) where R1= ^12u^0 JT W y + ^ 2u^0(ET W ) y; B1= T tr W2+ W0W + ^12b 0 X0 _J T W0W X b + ^ 2b 0 X0_(E T W0W ) X b h ^₁2X0 _J T W X b + ^ 2X0(ET W ) X b i0h X0b 1_Xi 1h_^ 2 1 X0 JT W X b + ^ 2X0(ET W ) X b i and (b; ^2₁; ^2) denote the restricted MLE under Hd

0: These are in fact the MLE under a random e¤ects

panel data model with no spatial lag dependence. ^u denotes the corresponding restricted MLE residuals under the null hypothesis Hd

0. This LM statistic is asymptotically distributed as 21 under H0d. ET = IT JT;

^2₁= ^u0_{P ^u=N; and ^}2

= ^u0_{Q^u=N (T 1): Note that LM}

= in (9) is of the same form as LM in (8). However,

R1 and B1are now di¤erent from R and B, and they are based on di¤erent restricted ML residuals, namely

^

u, those of a random e¤ects panel data model with no spatial lag dependence, see Baltagi (2005), rather than the OLS residuals ~u:

2.3

Conditional LM Test for

H

:

_{= 0 (assuming}

_{may or may not be zero)}

Similarly, if one uses LM de…ned in (8) to test Hc

0 : 2 = 0, one implicitly assumes that the spatial lag

dependence does not exist. This may lead to incorrect inference especially when is large. To overcome this problem, we derive a conditional LM test for no random region e¤ects given the existence of spatial lag dependence. The null hypothesis is He

0 : 2 = 0 (assuming may not be zero), and the conditional LM test

statistic is given by LM = = N T 2 (T 1)G 2 1; (10) where G1= Tu 0_{P u}

u0_u 1 and u denotes the restricted maximum likelihood residuals under the null

hypoth-esis H₀e; i.e., under a spatial lag dependence panel data model with no random e¤ects. Note that LM =

in (10) is of the same form as LM in (8). However, G1 di¤ers from G in that they are based on di¤erent

restricted ML residuals. The former is based on ut= yt W yt+ Xt ; where and are the MLE of and

in a spatial lag panel data model with no random e¤ects, while the latter is based on OLS residuals ~u: 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, Geographical Analysis, 20, 1–17.

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 Economet-rics. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York.

Baltagi, B. (2005), Econometric Analysis of Panel Data, New York, Wiley.

Baltagi, B.H., S.H. Song and W.Koh (2003) Testing Panel Data Regression Models with Spatial Error Correlation, Journal of Econometrics 117, 123–150.

Breusch, T.S. and A.R. Pagan (1980) The Lagrange multiplier test and its applications to model speci…cation in economet-rics, Review of Economic Studies 47, 239–253.

Kelejian, H.H. and I.R. Prucha (2001), On the asymptotic distribution of the Moran I test with applications. Journal of Econometrics 104, 219-257.

Ord, J. (1975), Estimation Methods for Models of Spatial Interaction, Journal of the American Statistical Association, 70, 120-126.

3

Appendix

3.1

The …rst-order and second-order derivatives

From the log-likelihood function given in (6), one can obtain the score equations as follows:

@L @ = T tr A 1_{W + u}0 1_(I T W ) y @L @ 2 = 1 2N T 2 1 + 1 2T 4 1 [u0P u] @L @ 2 = 1 2 N 2 1 + N (T 1) 2 + 1 2 u 0 4 1 P + 4Q u @L @ = X 0 1_u where 2

1= T 2 + 2; P = JT IN; JT = T 0T=T; with T denoting a vector of ones of dimension T:

The second-order derivatives are given by @2_L @ @ = T tr h W A 1 2i y0(IT W0) 1(IT W ) y @2_L @ @ 2 = T 4 1 u0P (IT W ) y @2_L @ @ 2 = u0 4 1 P + 4Q (IT W ) y @2_L @ @ = X 0 1_(I T W ) y = X0 12P + 2Q (IT W ) y @2L @ 2_@ 2 = 1 2N T 2 4 1 T2 6 1 [u0P u] @2_L @ 2_@ 2 = 1 2N T 4 1 T 16[u0P u] @2_L @ 2_@ = T X0 1_P 1_{u =} 4 1 T X0P u @2_L @ 2_@ 2 = 1 2 N 4 1 + N (T 1) 4 _u0 6 1 P + 6_{Q u} @2_L @ 2_@ = X0 1 1_{u = X}0 4 1 P + 4Q u @2L @ @ 0 = X 0 1_{X = X}0 2 1 P + 2Q X

3.2

Joint Test

Under the null hypothesis Ha

0 : = 2 = 0; equation (1) becomes a regression model with no spatial lag

dependence or random region e¤ects. The variance-covariance matrix reduces to 2_I

N T and the restricted

MLE of is ~_OLS, so that ~u = y X ~_OLS are the OLS residuals and ~2 = ~u0_{u=N T . This is clear from the~}

score equations evaluated under Ha

0 : = 2 = 0 : @L @ jH0a = T tr [W ] + ~ 2_u~0_(I T W ) y = ~ 2u~0(IT W ) y @L @ 2jH0a = 1 2N T ~ 2₊1 2T ~ 4_u~0_{P ~u =} 1 2N T T ~ u0_{P ~u} ~ u0_~u 1 ~ 2 @L @ 2jH0a = 1 2N T ~ 2₊1 2~ 4_u~0_{u = 0~} @L @ jH0a = ~ 2_X0_{u = 0~}

Therefore, the score with respect to 0 = ( ; 2_; 2_; 0_{), evaluated under the null hypothesis H}a

0 : =

~ D = 0 B B B B B B @ ~ D ~ D 2 ~ D 2 ~ D 1 C C C C C C A = 0 B B B B B B @ ~ 2u~0(IT W ) y N T 2~2 Tu~ 0_{P ~u} ~ u0_u~ 1 0 0 1 C C C C C C A = 0 B B B B B B @ R N T 2~2G 0 0 1 C C C C C C A

where R is a generalization of a similar term de…ned in Anselin (1988b) for the LM test of no spatial dependence in the cross-section case. In fact, R can be interpreted as N T times the regression coe¢ cient of (IT W ) y on ~u:

Under Ha

0, the elements of the information matrix ~J are given by:

E @ 2_L @ @ jHa0 = T tr W 2 ₊ e 2E [y0(IT W0W ) y] = T tr W2 +_e 2E [u0(IT W0W ) u] +e 2e 0 X0(IT W0W ) X e = T tr W2+ W0W +_e 2e0X0(IT W0W ) X e E @ 2_L @ @ 2 jHa0 = Te 4 E [u0P (IT W ) y] = Te 4E tr uu0 JT W = Te 2tr JT W = 0 E @ 2_L @ @ 2 jHa0 = e 4 E [u0(IT W ) y] =e 4E [tr (uu0(IT W ))] =e 2tr (IT W ) = 0 E @ 2_L @ @ jHa0 = e 2 E [X0(IT W ) y] =e 2X0(IT W ) X e E @ 2_L @ 2_@ 2 jHa0 = 1 2N T 2 e 4+ T2_e 6E [u0P u] = 1 2N T 2 e 4 E @ 2_L @ 2_@ 2 jHa0 = 1 2N Te 4 + T_e 6E [u0P u] = 1 2N Te 4 E @ 2_L @ 2_@ jHa0 = Te 4 E [X0P u] = 0 E @ 2_L @ 2_@ 2 jHa0 = 1 2N Te 4 +_e 6E [u0u] = 1 2N Te 4 E @ 2_L @ 2_@ jHa0 = e 4 E [X0u] = 0 E @ 2_L @ @ 0 jHa0 = e 2 X0X

Hence, the information matrix ~J evaluated under H0a can be written as

~ J = 0 @J~11 J~12 ~ J21 J~22 1 A 8

where ~J11= 0 @T tr W2+ W0W +e 2_e0 X0(IT W0W ) X e 0 0 1 2N T 2_e 4 1 A ; ~ J12= ~J210 = 0 @ 0 e 2 X0_(I T W ) X e 0 1 2N Te 4 0 1 A ; and ~J22= 0 @ 1 2N Te 4 0 0 _e 2X0_X 1 A :

Using partitioned inversion, we know that the upper 2 2 block of the inverse matrix ~J 1 _{is given by}

~

J11_{= J}~

11 J~12J~221J~21 1

: This can be easily derived as:

~ J11₌ 0 @B 1 0 0 _{N T (T}2 4 ₁₎ 1 A

where B = T tr W2+ W0W +_e 2e0X0(IT W0) M (IT W ) X e; and M = I X (X0X) 1X0. See

Anselin and Bera (1998) for a similar B term in the cross-section case. Therefore, the joint LM statistic for Ha

0 is given by LMJ = ~D0J~ 1D =~ D~ D~ 2 J~11 0 @D~ ~ D 2 1 A = R N T 2~2G 0 @B 1 0 0 _{N T (T}2 4 ₁₎ 1 A 0 @ R N T 2~2G 1 A = R2 B + N T 2(T 1)G 2_:

3.3

Conditional LM Test for

H

0

:

= 0 (assuming

> 0)

This section derives the conditional LM test for no spatial lag dependence given the existence of random region e¤ects. The null hypothesis is Hd

0 : = 0 (assuming 2 > 0). Under the null, the score equations are

given by @L @ jHd0 = T tr [W ] + ^u 0 _^ 2 1 P + ^ 2 Q (IT W ) y = ^12u^0 JT W y + ^ 2u^0(ET W ) y @L @ 2jHd 0 = 1 2N T ^ 2 1 + 1 2T ^ 4 1 [^u0P ^u] = 0 @L @ 2jHd 0 = 1 2 N ^ 2 1 + N (T 1) ^ 2 +1 2 ^u 0 _^ 4 1 P + ^ 4 Q ^u = 0 @L @ jHd0 = X 0 _^ 2 1 P + ^ 2Q ^u = 0

using tr [W ] = 0. Under the null hypothesis Hd

0, there is no spatial lag dependence and the

variance-covariance matrix = 2_J

T IN+ 2IN T. It is the familiar form of the one-way error component model,

see Baltagi (2005). The restricted MLE of ; 2_{; and} 2_{; are those based on MLE of a random e¤ects}

corresponding restricted MLE residuals are denoted by ^u: In fact, ^2₁= ^u0P ^u=N; and ^2 = ^u0Q^u=N (T 1): Therefore, the score with respect to 0= ( ; 2; 2; 0), evaluated under the null hypothesis H0d, is given by

^ D = 0 B B B B B B @ ^ D ^ D 2 ^ D 2 ^ D 1 C C C C C C A = 0 B B B B B B @ R1 0 0 0 1 C C C C C C A

where R1= ^12u^0 JT W y + ^ 2u^0(ET W ) y: Under H0d, the elements of the information matrix

b

J are given by:

E @ 2_L @ @ jH0d = T tr W 2 _{+ y}0_(I T W0) ^12P + ^ 2 Q (IT W ) y = T tr W2 + ^₁2E u0 JT W0W u + ^ 2E [u0(ET W0W ) u] +^₁2b0X0 JT W0W X b + ^ 2b 0 X0(ET W0W ) X b = T tr W2 + tr JT W0W + tr (ET W0W ) +^12b 0 X0 JT W0W X b + ^ 2b 0 X0(ET W0W ) X b = T tr W2+ W0W + ^₁2b0X0 JT W0W X b + ^ 2b 0 X0(ET W0W ) X b E @ 2_L @ @ 2 jH0d = Tb 4 1 E [u0P (IT W ) y] = Tb14E tr uu0 JT W = 0 E @ 2_L @ @ 2 jH0d = E h u0 _b14P +_b 4Q (IT W ) y i =_b12tr JT W +b 2tr (ET W ) = 0 E @ 2_L @ @ jH0d = E h X0 _b12P +_b 2Q (IT W ) y i =_b12X0 JT W X b +b 2X0(ET W ) X b E @ 2_L @ 2_@ 2 jHd0 = 1 2N T 2 b14+ T2b16E [u0P u] = 1 2N T 2 b14 E @ 2_L @ 2_@ 2 jHd 0 = 1 2N Tb 4 1 + Tb 6 1 E [u0P u] = 1 2N Tb 4 1 E @ 2_L @ 2_@ jHd 0 = Tb 4 1 E [X0P u] = 0 E @ 2_L @ 2_@ 2 jHd 0 = 1 2 h N_b14+ N (T 1)_b 4i+ Ehu0 _b16P +_b 6Q ui= 1 2 h N_b14+ N (T 1)_b 4i E @ 2_L @ 2_@ jHd 0 = E h X0 _b14P +_b 4Q ui= 0 E @ 2_L @ @ 0 jHd0 = X 0 _b 2 1 P +b 2 Q X

Therefore, the information matrix ^J evaluated under Hd

0 can be written as

^ J = 0 @J^11 J^12 ^ J21 J^22 1 A where ^J11= T tr W2+ W0W + ^₁2b0X0 JT W0W X b + ^ 2b 0 X0_{(ET W}0_{W ) X b ;} ^ J12= ^J210 = 0 0 b12X0 JT W X b +b 2X0(ET W ) X b 0 ; and ^J22= 0 B B B @ 1 2N T2b 4 1 12N Tb 4 1 0 1 2N Tb 4 1 12 h N_b14+ N (T 1)_b 4i 0 0 0 X0 _b 2 1 P +b 2 Q X 1 C C C A:

Using partitioned inversion, we know that the upper 1 1 element of the inverse matrix bJ 1 _{is given by}

b J11_{= J}^_{11 J}^_12J^ 1 22J^21 1 : Here ^ J11 J^12J^221J^21 = T tr W2_{+ W}0_{W + ^} 2 1 b 0 X0 _J T W0W X b + ^ 2b 0 X0_(E T W0W ) X b 0 0 _b12X0 _J T W X b +b 2X0(ET W ) X b 0 0 B B B @ 2b4 N T2_{(T 1)}+ 2b4 1 N T2 2b4 N T (T 1) 0 2b4 N T (T 1) 2b4 N (T 1) 0 0 0 hX0 _b 2 1 P +b 2Q X i 1 1 C C C A 0 B B B @ 0 0 b12X0 JT W X b +b 2X0(ET W ) X b 1 C C C A = T tr W2_{+ W}0_{W + ^} 2 1 b 0 X0 JT W0W X b + ^ 2b 0 X0(ET W0W ) X b h b 2 1 X0 JT W X b +b 2X0(ET W ) X b i0h X0_b 1_Xi 1h_b 2 1 X0 JT W X b +b 2X0(ET W ) X b i = bB1:

Therefore, the LM statistic for H0d is given by LM = = ^D0J^ 1D = R^ 1B11R1= R21=B1:

This is of the same form as LM for testing Hb

0 : = 0 (assuming no random e¤ects, i.e., 2 = 0).

However, R1 and B1 are now di¤erent from R and B. In fact, they are based on di¤erent restricted ML

residuals, namely ^u, those of a random e¤ects panel data model with no spatial lag dependence, see Baltagi (2005), rather than the OLS residuals ~u:

3.4

Conditional LM Test for

H

0

:

= 0 (assuming

may or may not be zero)

This section derives the conditional LM test for no random region e¤ects given the existence of spatial lag dependence. The null hypothesis is He

score equations are given by @L @ 2jH0e = 1 2N T 2₊1 2T 4_[u0_{P u]} @L @ jH0e = 2_X0_{u = 0} @L @ 2jH0e = 1 2N T 2₊1 2 4_u0_{u = 0} @L @ jH0e = T tr A 1_{W +} 2_u0_(I T W ) y = 0

Under the null hypothesis He

0, the variance-covariance matrix reduces to 2IN T and the restricted MLE of

and are in fact the MLE of a spatial lag model with no random e¤ects, see Anselin (1988a). These are denoted by and : Here, 2 = u0u=N T; with u = y (IT W ) y X : Therefore, the score with respect

to 0= ( 2_; 0_; 2_{; ), evaluated under the null hypothesis H}d

0, is given by D = 0 B B B B B B @ D 2 D D 2 D 1 C C C C C C A = 0 B B B B B B @ N T 2 2 Tu 0_{P u} u0_u 1 0 0 0 1 C C C C C C A = 0 B B B B B B @ N T 2 2G1 0 0 0 1 C C C C C C A 12

where G1= Tu

0_{P u}

u0_u 1 : Under H0e, the elements of the information matrix J are given by:

E @ 2_L @ 2_@ 2 jH0e = 1 2N T 2 4_{+ T}2 6_{E [u}0_{P u] =} 1 2N T 2 4 E @ 2_L @ 2_@ jH0e = T 4_{E [X}0_{P u] = 0} E @ 2_L @ 2_@ 2 jH0e = 1 2N T 4_{+ T} 6_{E [u}0_{P u] =} 1 2N T 4 E @ 2_L @ 2_@ jH0e = T 4_{E [u}0_{P (I} T W ) y] = T 4E tr uu0 JT W A 1 = T 2tr W A 1 E @ 2_L @ @ 0 jH0e = 2_X0_X E @ 2_L @ @ 2 jH0e = 4_{E [X}0_{u] = 0} E @ 2_L @ @ jH0e = 2_{E [X}0_(I T W ) y] = 2X0 IT W A 1 X E @ 2_L @ 2_@ 2 jH0e = 1 2N T 4₊ 6_{E [u}0_{u] =} 1 2N T 4 E @ 2_L @ 2_@ jH0e = 4_{E [u}0_(I T W ) y] = 4E tr uu0 IT W A 1 = T 2tr W A 1 E @ 2_L @ @ jH0e = T tr h W A 1 2i+ 2E [y0(IT W0W ) y] = T trh W A 1 2+ W A 1 0 W A 1 i+ 2 0X0 IT W A 1 0 W A 1 X

Therefore, the information matrix evaluated under H₀e can be written as: J = 0 @J11 J12 J21 J22 1 A with J11= 0 @ 1 2N T2 4 0 0 2X0_X 1 A ; J12= J210 = 0 @12N T 4 _T 2_{tr W A} 1 0 2_X0 _I T W A 1 X 0 1 A ; and J22= 0 @ 12N T 4 _T 2_{tr W A} 1 T 2tr W A 1 J 1 A ; where J = T trh W A 1 2_{+ W A} 1 0 _{W A} 1 i₊ 2 0_X0 _I T W A 1 0 W A 1 X : Using

partitioned inversion, we know that the upper 2 2 block of the inverse matrix J 1 _{is given by J}11 ₌

J11 J12J221J21 1

J11₌ 0 @ 2 4 N T (T 1) 0 0 _f 2_X0_X 1 H 1 2N T 4 2_X0 _I T W A 1 X 0 2X0 IT W A 1 X g 1 1 A : where H = 1 2N T 4 _{T tr}h _{W A} 1 2_{+ W A} 1 0 _{W A} 1 i₊ 2 0_X0 _I T W A 1 0 W A 1 X T 2_{tr W A} 1 2_:

We only need the …rst element of J11_{: Therefore, the LM statistic for H}e

0 is given by LM = = D0J 1D = D 2( 2 4 N T (T 1))D 2 = h N T 2 2G1 i2 ₂ 4 N T (T 1) = N T 2(T 1)G21:

This is of the same form as LM for testing Hc

0: 2 = 0 (assuming no spatial lag dependence, i.e., = 0).

However, G1= Tu

0_{P u}

u0_u 1 is based on di¤erent restricted ML residuals, ut= yt W yt+ Xt ; based on

the MLE of a spatial lag model with no random e¤ects, see Anselin (1988a), rather than the OLS residuals ~

u: