Instrumental Variable Estimation of a Spatial Autoregressive Panel Model with Random Effects

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1 Syracuse University SURFACE Center for Policy Research Maxwell School of Citizenship and Pulic Affairs Instrumental Variale Estimation of a Spatial Autoregressive Panel Model with Random Effects Badi. Baltagi Syracuse University Long Liu University of Texas at San Antonio Follow this and additional works at: Part of the Economics Commons Recommended Citation Baltagi, Badi. and Liu, Long, "Instrumental Variale Estimation of a Spatial Autoregressive Panel Model with Random Effects" (2011). Center for Policy Research. Paper This Working Paper is rought to you for free and open access y the Maxwell School of Citizenship and Pulic Affairs at SURFACE. It has een accepted for inclusion in Center for Policy Research y an authorized administrator of SURFACE. For more information, please contact surface@syr.edu.

2 ISSN: Center for Policy Research Working Paper No. 127 INSTRUMENTAL VARIABLE ESTIMATION OF A SPATIAL AUTOREGRESSIVE PANEL MODEL WIT RANDOM EFFECTS BADI. BALTAGI AND LONG LIU Center for Policy Research Maxwell School of Citizenship and Pulic Affairs Syracuse University 426 Eggers all Syracuse, New York (315) Fax (315) ctrpol@syr.edu January 2011 $5.00 Up-to-date information aout CPR s research projects and other activities is availale from our World Wide We site at www-cpr.maxwell.syr.edu. All recent working papers and Policy Briefs can e read and/or printed from there as well.

3 CENTER FOR POLICY RESEARC Spring 2011 Christine L. imes, Director Maxwell Professor of Sociology Associate Directors Margaret Austin Associate Director Budget and Administration Douglas Wolf Gerald B. Cramer Professor of Aging Studies Associate Director, Aging Studies Program John Yinger Professor of Economics and Pulic Administration Associate Director, Metropolitan Studies Program SENIOR RESEARC ASSOCIATES Badi Baltagi... Economics Roert Bifulco... Pulic Administration Leonard Burman.. Pulic Administration/Economics Kalena Cortes Education Thomas Dennison... Pulic Administration William Duncome... Pulic Administration Gary Engelhardt...Economics Madonna arrington Meyer... Sociology William C. orrace...economics Duke Kao... Economics Eric Kingson... Social Work Sharon Kioko..Pulic Administration Thomas Kniesner... Economics Jeffrey Kuik... Economics Andrew London... Sociology Len Lopoo... Pulic Administration Amy Lutz... Sociology Jerry Miner... Economics Jan Ondrich... Economics John Palmer... Pulic Administration David Popp... Pulic Administration Gretchen Purser... Sociology Christopher Rohlfs... Economics Stuart Rosenthal... Economics Ross Ruenstein... Pulic Administration Perry Singleton Economics Margaret Usdansky... Sociology Michael Wasylenko... Economics Jeffrey Weinstein Economics Janet Wilmoth... Sociology GRADUATE ASSOCIATES Charles Alamo... Pulic Administration Kanika Arora... Pulic Administration Samuel Brown... Pulic Administration Christian Buerger... Pulic Administration Il wan Chung... Pulic Administration Alissa Dunicki... Economics Andrew Friedson... Economics Virgilio Galdo... Economics Jenna arkaus... Pulic Administration Clorise arvey... Pulic Administration Biff Jones... Pulic Administration ee Seung Lee... Pulic Administration Chong Li... Economics Jing Li... Economics Allison Marier... Economics Qing Miao... Pulic Administration Wael Moussa... Economics Casey Muhm... Pulic Administration Kerri Raissian... Pulic Administration Morgan Romine... Pulic Administration Amanda Ross... Economics Natalee Simpson... Sociology Liu Tian... Pulic Administration Ryan Yeung... Pulic Administration STAFF Kelly Bogart Administrative Secretary Martha Bonney Pulications/Events Coordinator Karen Cimilluca...Office Coordinator Kitty Nasto Administrative Secretary Candi Patterson...Computer Consultant Roseann Presutti Administrative Secretary Mary Santy.... Administrative Secretary

4 Astract This paper extends the instrumental variale estimators of Kelejian and Prucha (1998) and Lee (2003) proposed for the cross-sectional spatial autoregressive model to the random effects spatial autoregressive panel data model. It also suggests an extension of the Baltagi (1981) error component 2SLS estimator to this spatial panel model. JEL No. C13, C21 Key Words: Panel Data; Spatial Model; Two Stage Least Squares; Error Components Address correspondence to: Badi. Baltagi, Center for Policy Research, 426 Eggers all, Syracuse University, Syracuse, NY ; Long Liu: Department of Economics, College of Business, University of Texas at San Antonio, One UTSA Circle, TX; ;

5 Instrument Variale Estimation of a Spatial Autoregressive Panel Model with Random E ects Badi. Baltagi, Long Liu y Decemer 22, 2010 Astract This paper extends the instrumental variale estimators of Kelejian and Prucha (1998) and Lee (2003) proposed for the cross-sectional spatial autoregressive model to the random e ects spatial autoregressive panel data model. It also suggests an extension of the Baltagi (1981) error component 2SLS estimator to this spatial panel model. Key Words: Panel Data; Spatial Model; Two Stage Least Squares; Error Components JEL: C13,C21 1 Model and Results Consider the following autoregressive spatial panel data model: y = W y + X + u; (1) u = Z + ; where y is of dimension NT 1, X is NT k, is k 1 and u is NT 1; see Anselin (1988), Anselin, Le Gallo and Jayet (2008), and Elhorst (2003) to mention a few. The oservations are ordered with t eing the slow running index and i the fast running index, i.e., y 0 = (y 11 ; : : : ; y 1N ; : : : ; y T 1 ; : : : ; y T N ) : X is assumed to e of full column rank and its elements are assumed to e asymptotically ounded in asolute value. The error component structure is given y the second equation with Z = T I N denoting the selector matrix for the (N 1) random vector of individual e ects which is assumed to e i.i.d. (0; 2 I N ). ere T is a vector of ones of dimension T and I N is an identity matrix of dimension N. is a vector of NT 1 remainder disturances which is assumed to e i.i.d. (0; 2 I NT ). Also, and are independent of each other and the regressor matrix X. Address correspondence to: Badi. Baltagi, Center for Policy Research, 426 Eggers all, Syracuse University, Syracuse, NY ; altagi@maxwell.syr.edu. y Long Liu: Department of Economics, College of Business, University of Texas at San Antonio, One UTSA Circle, TX ; long.liu@utsa.edu. 1

6 Ordering the data rst y time (with index t = 1; :::; T ) and then y individual units (with index i = 1; :::; N), we get W = I T W N, where the (N N) spatial weight matrix W N has zero diagonal elements and is row-normalized with its entries usually declining with distance. This in turn results in row and column sums of W N that are uniformly ounded in asolute value. In addition the column sums of W N, as well as the row and column sums of (I N W N ) 1 are ounded uniformly in asolute value y some nite constant. We also assume that is ounded in asolute value, i.e., jj < 1. This is a panel data version of the cross-section spatial autoregressive model considered y Kelejian, and Prucha (1998) and Lee (2003). Let A = I T A N where A N = I N W N, then one can write y = A 1 (X + u) : Note that E [W yu 0 ] = E W A 1 (X + u) u 0 = W A 1 6= 0; where = E (uu 0 ). The spatially lagged dependent variale W y is correlated with the disturance u. Therefore, the Ordinary Least Squares estimator will e inconsistent. Let Z = (X; W y) and = (; ) 0, the model in Equation (1) can e written as y = Z + u: (2) In the cross-section spatial autoregressive model, Kelejian, and Prucha (1998) suggested a two-stage least square estimator (2SLS) ased on feasile instruments like = X; W X; W 2 X which yield: ^2SLS = [Z 0 P Z] 1 Z 0 P y; (3) where P = ( 0 ) 1 0 denotes the projection matrix using. Actually, could include additional higher order terms like W 3 X; W 4 X, etc., see also Kelejian, Prucha and Yuzefovich (2004). Kelejian, and Prucha (1998) show that this 2SLS estimator is consistent under some general regularity conditions for this model. Note that the results derived in this paper carry through when includes these higher order terms. Let J T = J T =T; where J T is a matrix of ones of dimension T. Also, let E T = I T JT, and de ne P to e the projection matrix on Z ; i.e., P = J T I N, and Q = I NT P = E T I N. Premultiply Equation (2) y Q to otain ey = W ey + X e + eu: (4) This follows from the fact that QW = (E T I N )(I T W N ) = (E T W N ) = (I T W N )(E T I N ) = W Q: Applying the Kelejian, and Prucha (1998) 2SLS to this Q transformed panel autoregressive spatial model, we get the xed e ects spatial 2SLS (FE-S2SLS) estimator of ased upon e = ex; W X; e W 2 X e = 2

7 QX; QW X; QW 2 X = Q: We denote this y ^ F E S2SLS : Note that the s 2 of this FE-S2SLS provides a consistent estimate ^ 2 of 2 : Similarly, one could premultiply Equation (2) y P to otain y = W y + X + u: (5) This follows from the fact that P W = ( J T I N )(I T W N ) = ( J T W N ) = (I T W N )( J T I N ) = W P: Applying the Kelejian, and Prucha (1998) 2SLS to this P transformed panel autoregressive spatial model, we get the etween e ects spatial 2SLS (BE-S2SLS) estimator of ased upon = X; W X; W 2 X = P X; P W X; P W 2 X = P : We denote this y ^ BE S2SLS : Note that the s 2 of this BE-S2SLS provides a consistent estimate ^ 2 1 of 2 1 = T : As shown in Baltagi (2008), the variance-covariance matrix of u is = E (uu 0 ) = 2 1P + 2 Q, and 1=2 = 1 1 P + 1 Q. Left multiply Equation (2) y 1=2, we get y = Z + u ; (6) where y = 1=2 y, u = 1=2 u; and Z = 1=2 Z = 1=2 (X; W y) = X ; 1=2 W y = X ; W 1=2 y = (X ; W y ). The last expression follows from the fact that 1=2 W = 1 1 P W + 1 QW = 1 1 W P + 1 W Q = W 1=2 : This means that Equation (6) can also e written as: y = W y + X + u : (7) Applying the Kelejian, and Prucha (1998) 2SLS to this 1=2 transformed panel autoregressive spatial model, we get the random e ects spatial two-stage least square estimator (RE-S2SLS) of given y ^RE S2SLS = [Z 0 P Z ] 1 Z 0 P y (8) where = X ; W X ; W 2 X = 1=2 X; W 1=2 X; W 2 1=2 X = 1=2 X; 1=2 W X; 1=2 W 2 X = 1=2. Using the results in Baltagi (1981), one can similarly derive a spatial error component two-stage least square estimators (SEC-2SLS) as follows: Left multiply Equation (4) y ~ 0 ; and (5) y 0, and stack the two equation system recognizing that they estimate the same ; we get: ~ ey A ~ Z e A ~ 1 0 eu A ; 0 y 0 Z 0 u 0 where ~ eu A = 0 and V ~ eu A = 4 2 ~ 3 0 ~ 0 0 u 0 u This follows from the fact that Q = 2 Q; 1 0 and P = 2 1P; with QP = 0: Performing GLS on this two equation system we get the spatial error 3

8 component two-stage least squares (SEC-2SLS) estimator: ^SEC 2SLS = ez 0 P ~ e Z 2 + Z0 P Z 2 1! 1 ez 0 P ~ ey 2! + Z0 P y 2 1 (9) using the fact that Q and P are idempotent. = 2 Z 0 P ~ Z Z0 P Z 1 2 Z 0 P ~ y Z0 P y ; (10) Using a similar argument as in Baltagi (1981) one can show that ^ SEC 2SLS is a matrix weighted comination of ^ F E S2SLS and ^ BE S2SLS weighting each y the inverse of their variance-covariance matrix and such that the weights add up to the identity matrix. A feasile SEC-2SLS estimator can e otained y replacing ^ 2 and ^ 2 1 into Equation (9). Following a similar argument as in Cornwell, Schmidt and Wyhowski (1992), one can show that ^ SEC 2SLS can also e otained as 2SLS from the 1=2 transformed equation in (6) using B = ~; as instruments. To show this, note that P B = P ~ + P using the fact that ~ and are orthogonal to each other. This means that Z 0 P B Z = Z 0 P ~ Z + Z 0 P Z = 2 Z 0 P ~ Z Z0 P Z; since Q 1=2 = 1 Q; and P 1=2 = 1 1 P: Similarly, Z 0 P B y = Z 0 P ~ y + Z 0 P y = 2 Z 0 P ~ y Z0 P y. Therefore, ^ SEC 2SLS given y Equation (9) can e otained as 2SLS on (6) using B = ~; as instruments. Lee (2003) argued that in the cross-section spatial model, the optimal instruments for estimating in Equation (2) is E (Z) = E [X; W y] = X; W A 1 X : Therefore, a Lee (2003) type optimal instruments for estimating in the 1=2 transformed panel autoregressive spatial model in Equation (6) is E (Z ) = E 1=2 Z = E and the resulting est spatial 2SLS estimator is given y h i 1=2 (X; W y) = 1=2 X; 1=2 W A 1 X ; ^BS 2SLS = ( 0 Z ) 1 0 y ; (11) where = 1=2 X; 1=2 W A 1 X. A feasile version of this estimator is ased on consistent estimators of 2 1, 2, and respectively. Similarly, one can extend Baltagi s (1981) error component two-stage least squares estimator to this spatial autoregressive model using as instruments. Left multiply Equation (6) y ~ ; 0, where ~ = Q = 1 QX; QW A 1 X and = P = 1 1 P X; P W A 1 X, we get the two equation system recognizing that they estimate the same : ~ 1 0 0y A ~ 0 0 y Z 0Z 1 0 A ~ 0 u 0u 1 A ; 4

9 0 where ~ 1 0 0u A = 0 and V ~ 1 2 0u A = 4 ~ 0 ~ 3 0 0u 0u Performing GLS on this system we otain the spatial error component est two-stage least squares estimator (SEC-B2SLS): ^SEC B2SLS = = Z 0 P ~ Z + Z 0 P Z 1 Z 0 P ~ y + Z 0 P y (12) 1 2 Z 0 P ~ Z Z0 P Z 2 Z 0 P ~ y Z0 P y : (13) The second equality uses the fact that Q 1=2 = 1 Q and P 1=2 = 1 1 P: This SEC-B2SLS estimator can also e otained from the 1=2 transformed equation in (6) using B = ~ ; as instruments. In fact, ~ and are orthogonal to each other, since QP = 0: ence, P B = P ~ + P. This also implies that Z 0 P B Z = Z 0 P ~ Z + Z 0 P Z and Z 0 P B y = Z 0 P ~ y + Z 0 P y. Therefore, ^ SEC given y Equation (12) is the same as 2SLS on (6) using B = ~ ; as instruments. B2SLS Note that these 2SLS estimators are easy to apply using standard software. In fact, one can easily extend the ec2sls procedure in Stata for panels to the spatial panel case. Note also that SEC-2SLS and SEC-B2SLS use twice the instruments that their counterparts RE-S2SLS and BS-2SLS use. Although the set of instruments in the former is completely spanned y those for the latter estimators, these may yield smaller empirical standard errors in small samples, see Baltagi and Liu (2009) for a similar argument in the panel data case with no spatial correlation. REFERENCES Anselin, L. (1988), Spatial Econometrics: Methods and Models. Kluwer Academic Pulishers, Dordrecht. Anselin, L., J. Le Gallo and. Jayet (2008), Spatial Panel Econometrics. Ch. 19 in L. Mátyás and P. Sevestre, eds., The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice, Springer-Verlag, Berlin, Baltagi, B.. (1981), Simultaneous Equations with Error Components, Journal of Econometrics, 17, Baltagi, B.. (2008), Econometric Analysis of Panel Data, New York, Wiley. Baltagi, B.. and L. Liu (2009), A Note on the Application of EC2SLS and EC3SLS Estimators in Panel Data Models, Statistics and Proaility Letters, 79, Cornwell, C., P. Schmidt and D. Wyhowski (1992), Simultaneous Equations and Panel Data, Journal of Econometrics, 51, Elhorst, P.J. (2003), Speci cation and Estimation of Spatial Panel Data Models, International Regional Science Review, 26(3), Kelejian,.. and I.R.Prucha (1998), A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturances, Journal of Real Estate Finance and Economics, 17(1),

10 Kelejian,.., I.R.Prucha and Y. Yuzefovich (2004), Instrumental Variale Estimation of a Spatial Autoregressive Model with Autoregressive Disturances: Large and Small Sample Results, Advances in Econometrics, 18, Lee, L. (2003), Best Spatial Two-Stage Least Square Estimators for a Spatial Autoregressive Model with Autoregressive Disturances, Econometric Reviews, 22(4),