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

Transcription

1 Syracuse University SURFAC Center for Policy Researc Maxwell Scool of Citizensip and Public Affairs 8 Testing for Random ffects and Spatial Lag Dependence in Panel Data Models Badi H. Baltagi Syracuse University, bbaltagi@maxwell.syr.edu Long Liu Syracuse University Follow tis and additional works at: ttps://surface.syr.edu/cpr Part of te conometrics Commons Recommended Citation Baltagi, Badi H. and Liu, Long, "Testing for Random ffects and Spatial Lag Dependence in Panel Data Models" (8). Center for Policy Researc. 6. ttps://surface.syr.edu/cpr/6 Tis Working Paper is brougt to you for free and open access by te Maxwell Scool of Citizensip and Public Affairs at SURFAC. It as been accepted for inclusion in Center for Policy Researc by an autorized administrator of SURFAC. For more information, please contact surface@syr.edu.

2 ISSN: Center for Policy Researc Working Paper No. TSTING FOR RANDOM FFCTS AND SPATIAL LAG DPNDNC IN PANL DATA MODLS Badi H. Baltagi and Long Liu Center for Policy Researc Maxwell Scool of Citizensip and Public Affairs Syracuse University 6 ggers Hall Syracuse, New York 3- (35) 3-3 Fax (35) ctrpol@syr.edu Marc 8 $5. Up-to-date information about CPR s researc projects and oter activities is available from our World Wide Web site at www-cpr.maxwell.syr.edu. All recent working papers and Policy Briefs can be read and/or printed from tere as well.

3 CR FOR POLICY RSARCH Spring 8 Timoty Smeeding, Director Professor of conomics & Public Administration Associate Directors Margaret Austin Associate Director Budget and Administration Douglas Wolf Professor of Public Administration Associate Director, Aging Studies Program Jon Yinger Professor of conomics and Public Administration Associate Director, Metropolitan Studies Program SNIOR RSARCH ASSOCIATS Badi Baltagi... conomics Kalena Cortes ducation Tomas Dennison... Public Administration William Duncombe... Public Administration Gary ngelardt... conomics Debora Freund... Public Administration Madonna Harrington Meyer... Sociology Cristine Himes... Sociology William C. Horrace... conomics Duke Kao... conomics ric Kingson... Social Work Tomas Kniesner... conomics Jeffrey Kubik... conomics Andrew London... Sociology Len Lopoo... Public Administration Amy Lutz Sociology Jerry Miner... conomics Jan Ondric... conomics Jon Palmer... Public Administration Lori Ploutz-Snyder... xercise Science David Popp... Public Administration Cristoper Rolfs... conomics Stuart Rosental... conomics Ross Rubenstein... Public Administration Perry Singleton conomics Margaret Usdansky... Sociology Micael Wasylenko... conomics Janet Wilmot... Sociology GRADUAT ASSOCIATS Amy Agulay.Public Administration Javier Baez... conomics Sonali Ballal... Public Administration Jesse Bricker... conomics Maria Brown... Social Science Il Hwan Cung Public Administration Mike riksen... conomics Qu Feng... conomics Katie Fitzpatrick... conomics Cantell Frazier...Sociology Alexandre Genest... Public Administration Julie Anna Golebiewski...conomics Nadia Greenalg-Stanley... conomics Sung Hyo Hong... conomics Neelaksi Medi... Social Science Larry Miller... Public Administration Puong Nguyen... Public Administration Wendy Parker... Sociology Sawn Rolin... conomics Carrie Roseamelia... Sociology Jeff Tompson... conomics Coady Wing... Public Administration Ryan Yeung... Public Administration Can Zou... conomics STAFF Kelly Bogart Administrative Secretary Marta Bonney Publications/vents Coordinator Karen Cimilluca.... Administrative Secretary Roseann DiMarzo Receptionist/Office Coordinator Kitty Nasto..... Administrative Secretary Candi Patterson...Computer Consultant Mary Santy.... Administrative Secretary

4 Abstract Tis paper derives a joint Lagrange Multiplier (LM) test wic simultaneously tests for te absence of spatial lag dependence and random individual effects in a panel data regression model. It turns out tat tis LM statistic is te sum of two standard LM statistics. Te first one tests for te absence of spatial lag dependence ignoring te random individual effects, and te second one tests for te absence of random individual effects ignoring te spatial lag dependence. Tis paper also derives two conditional LM tests. Te first one tests for te absence of random individual effects witout ignoring te possible presence of spatial lag dependence. Te second one tests for te absence of spatial lag dependence witout ignoring te possible presence of random individual effects. JL codes: C and C3 Keywords: Panel Data; Spatial Lag Dependence; Lagrange Multiplier Tests; Random ffects.

5 Testing For Random ects and Spatial Lag Dependence in Panel Data Models Badi H. Baltagi, Long Liu y Syracuse University Marc 5, 8 Abstract Tis paper derives a joint Lagrange Multiplier (LM) test wic simultaneously tests for te absence of spatial lag dependence and random individual e ects in a panel data regression model. It turns out tat tis LM statistic is te sum of two standard LM statistics. Te rst one tests for te absence of spatial lag dependence ignoring te random individual e ects, and te second one tests for te absence of random individual e ects ignoring te spatial lag dependence. Tis paper also derives two conditional LM tests. Te rst one tests for te absence of random individual e ects witout ignoring te possible presence of spatial lag dependence. Te second one tests for te absence of spatial lag dependence witout ignoring te possible presence of random individual e ects. Key Words: Panel Data; Spatial Lag Dependence; Lagrange Multiplier Tests; Random ects. Introduction Spatial models deal wit correlation across spatial units usually in a cross-section setting, see Anselin (988a). Panel data models allow te researcer to control for eterogeneity across tese units, see Baltagi (5). Spatial panel models can control for bot eterogeneity and spatial correlation, see Baltagi, Song and Ko (3). Testing for spatial dependence as been extensively studied by Anselin (988a, 988b, ) and Anselin and Bera (998), to mention a few. Baltagi, Song and Ko (3) considered te problem of jointly testing for random region e ects in te panel as well as spatial correlation across tese regions. However, te last study allowed for spatial correlation only in te remainder error term. Tis paper generalizes te Baltagi, Song and Ko (3) to allow for spatial lag dependence of te autoregressive kind in te dependent variable rater tan te error term. In fact, tis paper derives a joint LM test wic simultaneously tests for te 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 Researc, 6 ggers Hall, Syracuse University, Syracuse, NY 3-; bbaltagi@maxwell.syr.edu. y Long Liu: conomics Department, ggers Hall, Syracuse University, Syracuse, NY 3-; loliu@maxwell.syr.edu.

6 turns out tat tis LM statistic is te sum of two standard LM statistics. Te rst LM, tests for te absence of spatial lag dependence ignoring te random individual e ects. Tis is te standard LM test derived in Anselin (988b) for cross-section data. Te second LM, tests for te absence of random individual e ects ignoring te spatial lag dependence. Tis is te standard LM test derived in Breusc and Pagan (98) for panel data. Tis paper also derives two conditional LM tests. Te rst one tests for te absence of random individual e ects witout ignoring te possible presence of spatial lag dependence. Te second one tests for te absence of spatial lag dependence witout ignoring te possible presence of random individual e ects. Tis sould provide useful diagnostics for applied researcers working in tis area. Te model and test statistics Consider a panel data regression model wit spatial lag dependence: y t W y t + X t + u t ; i ; : : : ; N; t ; :::; T () were yt (y t ; : : : ; y tn ) is a vector of observations on te dependent variables for N regions or ouseolds at time t ; :::; T: is a scalar spatial autoregressive coe cient and W is a known N N spatial weigt matrix wose diagonal elements are zero. W also satis es te condition tat (I N W ) is non-singular for all jj < : I N is an identity matrix of dimension N. X t is an N k matrix of observations on k explanatory variables at time t. u t (u t ; : : : ; u tn ) is a vector of disturbances following an error component model: u t + t () were ( ; : : : ; N ) and i is i.i.d. over i and is assumed to be N(; ): t ( t ; : : : ; tn ) and ti is i.i.d. over t and i and is assumed to be N(; ). Te f i g process is also independent of te f it g process. quation () can be rewritten in matrix notation as y (I T W ) y + X + u; i ; : : : ; N; t ; :::; T (3) were y is of dimension, X is k, is k and u is. Te observations are ordered wit t being te slow running index and i te fast running index, i.e., y (y ; : : : ; y N ; : : : ; y T ; : : : ; y T N ) : X is assumed to be of full column rank and its elements are assumed to be asymptotically bounded in absolute value. quation () can also be written in vector form as u ( T I N ) + ; ()

7 were ( ; : : : ; T ) ; T is a vector of ones of dimension T, I N is an identity matrix of dimension N; and denotes te Kronecker product. Under tese assumptions, te variance-covariance matrix for u can be written as were J T is a matrix of ones of dimension T. (J T I N ) + (I T I N ) ; (5) Under te normality assumption, te log-likeliood function of equation () is given by L ln ln jj + T ln jaj [(I T A) y X] [(I T A) y X] (6) were A I N W: Ord (975) sows tat ln ji N W j P N i ln (! i), were! i s are te eigenvalues of W. Using te notation in Baltagi (5), we can write ; were Q+ P; P J T I N ; J T T T T; Q I T N P; and T +. From wic it follows tat ln jj ln +N ln. Te log-likeliood function in (6) can be rewritten as L ln ln + N ln +T NX ln (! i ) and one can estimate tis model using maximum likeliood, see Anselin (988a). i [(I T A) y X] [(I T A) y X] Tis paper derives a joint LM test for te absence of spatial lag dependence as well as random e ects. Te null ypotesis is H a : ; and te alternative H a is tat at least one component is not zero. Tis generalizes te LM test derived in Anselin (988b) for te absence of spatial lag dependence H b : (assuming no random e ects, i.e., ), and te Breusc and Pagan (98) LM test for te absence of random e ects H c : (assuming no spatial lag dependence, i.e., ). We also derive two conditional LM tests, one for H d : (assuming te possible existence of random e ects, i.e., > ); and te oter one for H e : (assuming te possible existence of spatial lag dependence, i.e., may be di erent from zero). All te proofs are given in te Appendix to te paper. (7). Joint LM test for H a : Te joint LM test statistic for testing H a : is given by LM J R B + (T ) G LM + LM (8) 3

8 were B T tr W + W W +e e X (I T W ) M (I T W ) X e ; M I X (X X) X ; G T ~u P ~u ~u ~u ; R ~u (I T W )y ~u ~u : LM R B; and LM G (T ) : e is te restricted ML under H a wic yields OLS, ~u denotes te OLS residuals, and e ~u ~u. R is a generalization of a similar term de ned in Anselin (988b) for te LM test of no spatial dependence in te cross-section case. In fact, R can be interpreted as times te regression coe cient of (I T W ) y on ~u: Here, te joint LM test LM J te sum of two LM test statistics: Te rst is LM R B;wic is te LM test statistic for testing H b : assuming tere is no random region e ects, i.e., assuming, see Anselin (988a). LM is asymptotically distributed as under H b : Te second is LM (T ) G ;wic is te LM test statistic for testing H c : assuming tere is no spatial lag dependence, i.e., assuming tat, see Breusc and Pagan (98). Since LM and LM are asymptotically independent, LM J is asymptotically distributed as under H a. It is important to point out tat te asymptotic distribution of our test statistics are not explicitly derived in te paper but tat tey are likely to old under a similar set of primitive assumptions developed by Kelejian and Pruca (). is. Conditional LM Test for H d : (assuming > ) Wen one uses LM de ned in (8) to test H b :, one implicitly assumes tat te random region e ects do not exist. Tis may lead to incorrect inference especially wen is large. To overcome tis problem, we derive a conditional LM test for no spatial lag dependence assuming te possible existence of random region e ects. Te null ypotesis is H d : (assuming > ), and te conditional LM test statistic is given by LM R B ; (9) were R ^ ^u JT W y + ^ ^u ( T W ) y; B T tr W + W W + ^ b X JT W W X b + ^ b X ( T W W ) X b ^ X JT W X b + ^ X ( T W ) X b i i X b X ^ X JT W X b + ^ X ( T W ) X b i and ( b ; ^ ; ^ ) denote te restricted ML under H d : Tese are in fact te ML under a random e ects panel data model wit no spatial lag dependence. ^u denotes te corresponding restricted ML residuals under te null ypotesis H d. Tis LM statistic is asymptotically distributed as under H d. T I T JT ; ^ ^u P ^un; and ^ ^u Q^uN(T ): Note tat LM in (9) is of te same form as LM in (8). However, R and B are now di erent from R and B, and tey are based on di erent restricted ML residuals, namely

9 ^u, tose of a random e ects panel data model wit no spatial lag dependence, see Baltagi (5), rater tan te OLS residuals ~u:.3 Conditional LM Test for H e : (assuming may or may not be zero) Similarly, if one uses LM de ned in (8) to test H c :, one implicitly assumes tat te spatial lag dependence does not exist. Tis may lead to incorrect inference especially wen is large. To overcome tis problem, we derive a conditional LM test for no random region e ects given te existence of spatial lag dependence. Te null ypotesis is H e : (assuming may not be zero), and te conditional LM test statistic is given by LM (T ) G ; () were G T u P u u u and u denotes te restricted maximum likeliood residuals under te null ypotesis H; e i.e., under a spatial lag dependence panel data model wit no random e ects. Note tat LM in () is of te same form as LM in (8). However, G di ers from G in tat tey are based on di erent restricted ML residuals. Te former is based on u t y t W y t + X t ; were and are te ML of and in a spatial lag panel data model wit no random e ects, wile te latter is based on OLS residuals ~u: RFRNCS Anselin, L. (988a) Spatial conometrics: Metods and Models. Kluwer Academic Publisers, Dordrect. Anselin, L. (988b) Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity, Geograpical Analysis,, 7. Anselin, L. () Rao s score tests in spatial econometrics. Journal of Statistical Planning and Inference, 97, Anselin, L., Bera, A.K. (998) Spatial Dependence in Linear Regression Models wit an Introduction to Spatial conometrics. In: Ulla, A., Giles, D..A. (ds.), Handbook of Applied conomic Statistics. Marcel Dekker, New York. Baltagi, B. (5), conometric Analysis of Panel Data, New York, Wiley. Baltagi, B.H., S.H. Song and W.Ko (3) Testing Panel Data Regression Models wit Spatial rror Correlation, Journal of conometrics 7, 3 5. Breusc, T.S. and A.R. Pagan (98) Te Lagrange multiplier test and its applications to model speci cation in econometrics, Review of conomic Studies 7, Kelejian, H.H. and I.R. Pruca (), On te asymptotic distribution of te Moran I test wit applications. Journal of conometrics,

10 Ord, J. (975), stimation Metods for Models of Spatial Interaction, Journal of te American Statistical Association, 7, Appendix 3. Te rst-order and second-order derivatives From te log-likeliood function given in (6), one can obtain te score equations T tr A W + u (I T W ) y + T [u P u] X u N + N (T ) + u P + Q u were T + ; P J T I N ; JT T T T; wit T denoting a vector of ones of dimension T: 6

11 Te second-order derivatives are given by 3. Joint @ Under te null ypotesis H a T tr W A i T u P (I T W ) y u P + Q (I T W ) y y (I T W ) (I T W ) y X (I T W ) y X P + Q (I T W ) y T 6 [u P u] T 6 [u P u] T X P u T X P u N + N (T ) u 6 X u X P + Q X X X P + Q X P + 6 Q u : ; equation () becomes a regression model wit no spatial lag dependence or random region e ects. Te variance-covariance matrix reduces to I and te restricted ML of is ~ OLS, so tat ~u y score equations evaluated under H a : : X ~ OLS are te OLS residuals and ~ ~u ~u. Tis is clear from T tr [W ] + ~ ~u (I T W ) y ~ ~u (I T W ) y ~ + T ~ ~u P ~u T ~u P ~u @ ~ X ~u ~ + ~ ~u ~u Terefore, te score wit respect to (; ; ; ), evaluated under te null ypotesis H a : is given by ~ 7

12 ~D ~ ~u (I T W ) y R ~D ~D T ~u P ~u ~ ~u ~u G ~ B C B C B C A ~D were R is a generalization of a similar term de ned in Anselin (988b) for te LM test of no spatial dependence in te cross-section case. In fact, R can be interpreted as times te regression coe cient of (I T W ) y on ~u: Under H a, te elements of te information matrix ~ J are given T tr W + e [y (I T W W @@ T tr W + e [u (I T W W ) u] + e e X (I T W W ) X e T tr W + W W + e e X (I T W W ) X e T e [u P (I T W ) y] T e tr uu JT W T e tr J T W e [u (I T W ) y] e [tr (uu (I T W ))] e e [X (I T W ) y] e X (I T W ) @ T e e tr (I T W ) e + T e 6 [u P u] e e + T e 6 [u P u] e [X P u] e + e 6 [u u] e [X u] e X X Hence, te information matrix J ~ evaluated under H a can be written as ~ J J ~ A ~J J ~ 8

13 were ~ T tr W + W W + e e X (I T W W ) X e A ; e ~J J e X (I T W ) X e A ; and J e A : e e X X Using partitioned inversion, we know tat te upper block of te inverse matrix J ~ is given by ~J ~J ~ J ~ J ~ J : Tis can be easily derived as: B A (T ) were B T tr W + W W + e e X (I T W ) M (I T W ) X e ; and M I X (X X) X. See Anselin and Bera (998) for a similar B term in te cross-section case. Terefore, te joint LM statistic for H a is given by LM J D ~ J ~ D ~ ~D D ~ ~ D A R ~D R B + (T ) G : B ~ (T ) R ~ G A 3.3 Conditional LM Test for H d : (assuming > ) Tis section derives te conditional LM test for no spatial lag dependence given te existence of random region e ects. Te null ypotesis is H d : (assuming > ). Under te null, te score equations are given j @ using tr [W ]. T tr [W ] + ^u ^ P + ^ Q (I T W ) y ^ ^ + T ^ [^u P ^u] N ^ + N (T ) ^ + ^u X ^ P + ^ Q ^u ^u JT W y + ^ ^u ( T W ) y ^ P + ^ Q ^u Under te null ypotesis H d, tere is no spatial lag dependence and te variancecovariance matrix J T I N + I. It is te familiar form of te one-way error component model, see Baltagi (5). Te restricted ML of ; ; and ; are tose based on ML of a random e ects panel data model wit no spatial lag dependence. Tese are denoted by b ; b ; and b ; respectively. Te 9

14 corresponding restricted ML residuals are denoted by ^u: In fact, ^ ^u P ^un; and ^ ^u Q^uN(T ): Terefore, te score wit respect to (; ; ; ), evaluated under te null ypotesis H d, is given by ^D R ^D ^D B C B C A ^D were R ^ ^u JT W y + ^ ^u ( T W ) y: Under H d, te elements of te information matrix bj are given L H d T tr W + y (I T W ) ^ P + ^ Q (I T W ) y T tr W + ^ u JT W W u + ^ [u ( T W W ) u] +^ b X JT W W X b + ^ b X ( T W W ) X b T tr W + tr J T W W + tr ( T W W ) +^ b X JT W W X b + ^ b X ( T W W ) X b T tr W + W W + ^ b X JT W W X b + ^ b X ( T W W ) X T b [u P (I T W ) y] T b tr uu JT L i u b P + b Q (I T W ) y b tr J T W + b tr ( T @ X b T b P + b i Q (I T W ) y b b + T b 6 [u P u] b b + T b 6 [u P u] b [X P u] i N b + N (T ) b + u b 6 X b X b P + b P + b Q X i Q u X JT W X b + b X ( T W ) X b P + b 6 Terefore, te information matrix ^J evaluated under H d can be written as i Q u i N b + N (T ) b

15 ^J ^J ^J ^J A were ^J T tr W + W W + ^ b X JT W W X b + ^ ^J ^J b X JT W X b + b X ( T W ) X b ; and ^J b b i b N b + N (T ) b X b P + b b X ( T W W ) X b Q C A : X Using partitioned inversion, we know tat te upper element of te inverse matrix J b is given by bj ^J ^J ^J ^J : Here ^J ^J ^J ^J T tr W + W W + ^ b X JT W W X b + ^ b X ( T W W ) X b b X JT W X b + b X ( T W ) X b b (T ) + b b (T ) b (T ) b N(T ) B i A X b P + b Q X B b X JT W X b + b X ( T W ) X b A T tr W + W W + ^ b X JT W W X b + ^ b X JT W X b + b X ( T W ) X b i X b X b X ( T W W ) X b i b X JT W X b + b ; X ( T W ) X b i b B : Terefore, te LM statistic for H d is given by LM ^D ^J ^D R B R R B : Tis is of te same form as LM for testing H b : (assuming no random e ects, i.e., ). However, R and B are now di erent from R and B. In fact, tey are based on di erent restricted ML residuals, namely ^u, tose of a random e ects panel data model wit no spatial lag dependence, see Baltagi (5), rater tan te OLS residuals ~u: 3. Conditional LM Test for H e : (assuming may or may not be zero) Tis section derives te conditional LM test for no random region e ects given te existence of spatial lag dependence. Te null ypotesis is H e : (assuming tat may not be zero). Under te null, te

16 score equations are given + T [u P X u + u T tr A W + u (I T W ) Under te null ypotesis H e, te variance-covariance matrix reduces to I and te restricted ML of and are in fact te ML of a spatial lag model wit no random e ects, see Anselin (988a). Tese are denoted by and : Here, u u; wit u y (I T W ) y X: Terefore, te score wit respect to ( ; ; ; ), evaluated under te null ypotesis H d, is given by D T u P u u u G D D B C B C B C A D

17 were G T u P u u u : Under H, e te elements of te information matrix J are T + T 6 [u P u] [X P u] + T 6 [u P u] T [u P (I T W ) y] T X X [X u] [X (I T W ) y] + 6 [u u] [u (I T W ) y] T tr T tr tr uu JT W A T tr W A X I T W A X tr uu I T W A T tr W A W A i + [y (I T W W ) y] W A + W A W A i + X I T W A Terefore, te information matrix evaluated under H e can be written as: W A X J J A J J wit A ; X X J T tr W A X I T W A X A ; and T tr W A T tr W A A ; J were J T tr W A + W A W A i + X I T W A W A X: Using partitioned inversion, we know tat te upper block of te inverse matrix J is given by J J J J J : After some tedious algebra, tis can be derived as: 3

18 (T ) f X X H were H T tr W A + W A T tr W A : X I T W A X X I T W A X g W A i + X I T W A We only need te rst element of J : Terefore, te LM statistic for H e is given by LM D J D D ( (T ) ) D i G (T ) (T ) G : A : W A X Tis is of te same form as LM for testing H c : (assuming no spatial lag dependence, i.e., ). However, G T u P u u u is based on di erent restricted ML residuals, u t y t W y t + X t ; based on te ML of a spatial lag model wit no random e ects, see Anselin (988a), rater tan te OLS residuals ~u: