Performance of the FGS3SLS Estimator in Small Samples: A Monte Carlo Study

Transcription

1 The Regonal Economcs Applcatons Laboratory (REAL) s a unt n the Unversty of Illnos focusng on the development and use of analytcal models for urban and regon economc development. The purpose of the Dscusson Papers s to crculate ntermedate and fnal results of ths research among readers wthn and outsde REAL. The opnons and conclusons expressed n the papers are those of the authors and do not necessarly represent those of the Unversty of Illnos. All requests and comments should be drected to Geoffrey J. D. Hewngs, Drector, Regonal Economcs Applcatons Laboratory, 67 South Mathews, Urbana, IL, , phone (7) , FAX (7) Web page: Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study Saket Sarraf, Keran P. Donaghy, Geoffrey J.D. Hewngs REAL 3-T- May, 3

2 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study Saket Sarraf Afflate Research Assstant Professor, Regonal Economcs Applcatons Laboratory, Unversty of Illnos, Urbana, IL 68, USA, and Prncpal, ps Collectve, A-5, Galaxy Tower, Nr Hotel Grand Bhagwat, Bodakdev, Ahmedabad 3854, Inda saket.sarraf@gmal.com Keran P. Donaghy Department of Cty and Regonal Plannng, Cornell Unversty, Ithaca, NY, 4853 kpd3@cornell.edu Geoffrey Hewngs Regonal Economcs Applcatons Laboratory, Unversty of Illnos, Urbana, IL 68, USA hewngs@llnos.edu Abstract: System of equatons models wth spatal lags n dependent varable and error terms can be estmated usng the full nformaton Feasble Generalzed Spatal Three Stages Least Square (FGS3SLS) estmator proposed by Kelejan & Prucha (4). The estmator s consstent and asymptotcally normal, but ts fnte sample propertes are not analytcally determnable. In absence of very large samples as s the case n most appled work, t s dffcult to nterpret the results wth confdence based on asymptotc results only. Ths paper evaluates the performance of the FGS3SLS estmator n fnte samples and ts senstvty to varyng degrees of spatal nteracton and externaltes usng Monte Carlo smulatons. Key words: FGS3SLS, Spatal nteracton models, Monte Carlo smulaton, Fnte sample propertes JEL Classfcaton: C3, C, C3, R4. Introducton The modelng of spatal processes has attaned a manstream poston n socal scences (Goodchld et al., ). Anseln () presents an hstorcal analyss of how spatal econometrcs has attaned a manstream status n appled econometrcs and socal scence methodology. In the smplest cases, the varables of nterest are spatally correlated wth ther neghbors and wth other varables. As we move from one varable to a system of varables, modelng the spatal nteractons becomes complex. The complexty further ncreases as the

3 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 3 randomness becomes correlated spatally and across equatons. Modelng the strength of spatal nteractons and externaltes requres the specfcaton and estmaton of spatal econometrc models. However, the avalable estmators (Anseln, 988; Case, 99; Case et al., 993) lack methodologcal sophstcaton and computatonal smplcty to accurately estmate smultaneous systems wth spatal autoregressve dependent varables and spatally nterrelated cross sectonal equatons. They are often based on quas-maxmum lkelhood procedures and mght not have feasble solutons n medum to large samples. Further, they are desgned for sngle equaton frameworks (See Kelejan & Prucha, 999 for an extensve dscusson on ths ssue). To estmate models for such processes, Kelejan & Prucha (4) proposed the lmted nformaton Feasble Generalzed Spatal Two Stage (FGSSLS) and full nformaton Feasble Generalzed Three Stage Estmators (FGS3SLS). These estmators are based on generalzed methods of moments usng approxmaton of optmal nstruments, and thus are computatonally smple. Kelejan and Prucha show that the estmators are consstent and asymptotcally normal. Some of the appled examples of ths estmator nclude Ngeleza et al. (6) to determne the geographcal and nsttutonal determnants of real ncome, Drffeld (6) for modelng spatal spllovers of foregn drect nvestment, Fshback et al. (6) for modelng the mpact of New Deal expendtures on moblty durng the great depresson. More recent applcatons ncludes applcatons n the felds of assessng regonal growth (Gebremaram et al. ).. It s mportant to understand how ths estmator behaves n appled studes gven ts relevance n estmatng many of the complex spatal processes that have been largely gnored thus far. However, our understandng of ths estmator s at best, rudmentary. The number of publcatons usng ths estmator s relatvely few, and only ts asymptotc propertes have been establshed so far. In absence of very large samples, as s the case n much appled work, t s dffcult to nterpret the results wth confdence based on asymptotc results only. One alternatve to employ n a stuaton such as ths s to use fnte sample approxmatons or asymptotc expansons. However, these approxmatons tend to be very complex, the results dffcult to nterpret and the computatons very advanced. Some early work on ths topc s summarzed n Phlps (983) and Rothenberg (984). In contrast, the method of Monte Carlo replaces the sklls needed n asymptotc approxmatons by relyng on computatonal power of

4 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 4 computers. In ths paper, the propertes of the parameters of nterest are studed through a seres of stochastc smulatons and ther statstcs are analyzed (Davdson & MacKnnon, 993). Ths paper nvestgates the performance of the FGS3SLS estmator for a system of smultaneous equatons, wth spatal autoregressve dependent varables and spatally autocorrelated error structures usng Monte Carlo experments. Performance s measured by ts ablty to estmate parameters of the model and senstvty of the results to varyng degree of spatal dependences, choce of spatal weght matrx, sample sze and varance covarance matrces. The paper concludes by emphaszng the need for further studes on the subject to ncrease our understandng of the estmator s behavor n appled work. The rest of the paper s organzed as follows. Secton sets up the model used for the study. Secton 3 brefly descrbes the estmator and secton 4 descrbes the expermental desgn. The results of the smulaton exercse are presented n secton 5. Secton 6 summarzes the man fndngs and concludes the study wth drecton for future research.. Model Structure. Formal Consderatons The performance of the FGS3SLS estmator was tested usng a model specfcaton closely resemblng the structure of the model used n Sarraf () to analyze the regonal socal dynamcs and ts mpacts on land-use change. The model used here conssts of a system of smultaneous equatons wth two endogenous varables, ther spatal and temporal lags and two exogenous varables. The spatal lag of the dependent varable s treated as endogenous whle the temporal lag s consdered as predetermned, snce the model s condtoned on past values of the dependent varable. The dsturbances are assumed to be correlated across space and across dfferent equatons. Ths form of model allows the analyst (a) to capture spatal processes lke dffuson across space, (b) to address problems of ecologcal fallacy or presence of some local condtons leadng to spatally correlated error structures, and (c) to determne the correlaton between two spatal processes. Further, the specfcaton allows forecastng of the value of dependent varables condtonal on ts own past values, and other exogenous varables after accountng for the underlyng spatal processes.

5 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 5 Let y represent percent abandoned housng unts n a census tract and y represent net nmgraton of households. Equaton states that percent abandoned unts y depend on: () the magntude of net n-mgraton of households ( y ) and percent housng abandonment n neghborng tracts ( Wy ) n the current perod; () the percentage of the housng abandoned n the prevous perod ( x ); (3) the dstance from nterstate ( x 3 ); and (4) a random component ( u ). Smultaneously, the net n-mgraton of households s endogenous and depends on the percentage of unts abandoned snce hgher housng abandonment tends to repel more households from the regon. Accordng to the equaton (), the magntude of net n mgraton of households ( y ) n a tract depends on: () the percentage of housng abandonment ( y ) and net n mgraton of households n the neghborng tracts ( Wy ) n the current perod; () lagged values of net nmgraton ( x ); (3) the condton of nfrastructure ( x 4 ); and (4) a random component u. Thus, housng abandonment and net n mgraton of households are jontly determned. Note that x 3 s treated as fxed over tme whle x 4 s tme dependent but stll exogenous. y y W y x x u () 3 3 y y W y x x u () 4 4 where, y,(,) represents the endogenous varables we are nterested to forecast. Wy s are the spatally lagged dependent varables wth the spatal lag parameter. W s the row standardzed weght matrx of known constants descrbng the neghborhood structure of observatons. x and xare the temporally lagged values of dependent varable y and y respectvely, x 3 and x 4 are the exogenously determned varables whose values ether reman fxed throughout the smulaton or are known a pror. u and u represent the stochastc component of the model whose behavor s elaborated below. The dsturbance vectors u and u n equatons () and () are assumed to be correlated across space and across equatons. The spatal geography over whch the socal dynamcs are occurrng s dfferent from the admnstratve geography of census tracts. The aggregaton of data at the tract level leads to correlaton of dsturbances across tracts. Further any randomness affectng

6 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 6 housng abandonment and change n number of households may be correlated. Thus, the current specfcaton allows for randomness that s also correlated across equatons. u u W u (3) 3 W u (4) 4 wth, Cov(, ) (5) Equatons (3) and (4) characterze the correlaton across space where Wu 3 and Wu 4 are average values of error terms n the neghborng locatons, and and depct the degree of spatal correlaton of the error terms. and are non-spatally correlated dsturbances but are correlated across equatons wth the varance covarance matrx (equaton 5). Ths completes the specfcaton of the hypothetcal model.. Generalzed form For brevty, the model system represented n equatons () to (5) can be rewrtten n matrx notaton as: In. W. In y. In. In x 3. In. In x3. In In. W y. In. In x. In 4. In x4 ( W 3). ( W4 ). (6) or, BY. T. X T. X U (7) a a b b where, Y y, y s a vector of endogenous varables, X x, x lagged endogenous varable Y, X x, x b 3 4 a s a vector of temporally s a vector of exogenous varables and In s an dentty matrx of dmenson n. B, T a and T b represent the coeffcents assocated wth these varables n equaton (6). U u, u s descrbed n Appendx C. represents the vector of dsturbance terms. The estmator

7 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 7 3. Monte Carlo Experments Wth the model structure n place, desgnng the Monte Carlo experment conssts of three addtonal parts, namely: defnng the parameter settngs; generatng the spato-temporal array of synthetc data for dfferent varables consstent wth the underlyng spatal process; and desgnng the smulatons to reduce errors due to randomzaton and analyss of alternatve scenaros. Each of these steps s elaborated below. 3. Parameter settngs Ths secton assgns values to the parameters used n the model specfed n equatons () through (5) ncludng the values of all the coeffcents, the varance covarance matrx of dsturbance terms, the weght matrx and the spatal dependence parameters. The parameters for the spatal lag of the dependent varable and for spatal autocorrelaton n the error terms {, } nclude all possble combnatons from the set {-.8,, -.4, -.,,.,.4,.6,.8} n dfferent experments for each choce of. For clarty of the exposton, we assume a common neghborhood structure W( W W W3 W4 ), and. It should be noted that n applcatons, ths s not the case. Weght matrces for dfferent varables wll take dfferent specfcatons dependng on the nature of spatal processes that nfluence them (see for example, Cuaresma, ). However, there s no loss of generalty by usng the same weght matrx for dfferent process for the Monte Carlo experments. We consder three samples szes of, 5 and 5 observatons each. For each sample sze, two dfferent weght matrces are consdered. The specfcaton of W closely follows the weght matrx descrbed n Kelejan & Prucha (999) and Das, Kelejan, & Prucha (3). These matrces dffer n the degree of sparseness. For the frst specfcaton, a hypothetcal crcular world s consdered where each observaton ( y andu ) s related to exactly one neghbor mmedately before t and one neghbor mmedately after t. Thus, the th row of W has non-zero enttes only n - and + column, for each =, 3. (n-). For the frst row, the non-zero elements are n the nd and n th column whle for the last row, the non-zero elements are n the n- th and st column. Further, the matrx W s row standardzed such that sum of elements n each row =. Ths matrx s termed as one ahead and one behnd. The second matrx s analogously defned as three ahead and three behnd where each observaton s related to exactly three

8 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 8 other observatons behnd and ahead of t. Thus, the average number of neghbors n the frst matrx s whle n the second matrx t s 6. Kelejan and Prucha report that the results from hypothetcal weght matrces and real world weght matrces are smlar. We conjecture smlar outcomes n ths case. The parameter assocated wth the non-spatal components of the model s specfed as below, representng both postve and negatve assocaton. The choce of these values has no or very lttle bearng on the research queston..3,.7,.,.5.5,. 3 4 Smlarly, two alternatve forms of the varance covarance matrx are used correspondng to an R value of roughly.75 and.6 respectvely, where R s defned as the average squared correlaton coeffcent (Carter & Nagar, 977) between y and the mean value of y as explaned by the model n dfferent experments: = and = In the frst case, and the correlaton between error terms corr(, ) / s.5. In the second case, and the correlaton between error terms s Generatng Synthetc Data A dataset that s a realzaton of the spatal processes under study s needed for the purpose of estmaton. It should be a generated from nterdependences between varables, random components and the spatal nteractons between them as specfed n the model structure. For each scenaro, a dfferent dataset s generated nfluenced by the parameter settngs, nature and strength of spatal dependence, varance-covarance structure and sample sze. Ths procedure wll ensure that the varaton n results of dfferent scenaros only reflects the changes n the scenaros rather than the randomness n the data generaton process thus makng comparson possble. The data generaton process conssts of two parts, namely generatng the values of the dsturbance terms and that for the regresson varables. 3.3 Generatng the values of dsturbance terms

9 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 9 The process of generatng spatally correlated random components starts wth random draws from ndependently and dentcally dstrbuted normal random varables,(, ) wth zero mean and unt varance. These values are then transformed to reduced form dsturbances that are correlated across equatons wth zero mean and varance covarance matrx usng the followng transformaton: E V * where, E (, ), V (, ) and * s the m x m lower trangular matrx such that. The dsturbance terms, u, n ' * * the model are then obtaned by usng the transformaton randomness that s correlated across space as well as across equatons. u I W ( ) resultng n 3.4 Generatng the values of regresson varables The startng values for a large number of tme seres for the two exogenous varables X bt, 3 4 x, x are ndependently drawn from a normal dstrbuton wth zero mean and unt varance. x 3 s treated as fxed over tme whle x 4 s assumed to grow at a rate of one percent n every perod. To avod the senstvty of results to exogenous varables, they are generated usng the same set of random realzatons n every experment. Values of Y t are generated condtonal on X at, and X bt, usng a reduced form autoregressve data generaton process descrbed as follows. Re-wrtng equaton (7) wth a tme subscrpt, substtutng Xa, t Yt and takng expectatons, we obtan: BY. T. Y T. X (8) t a t b b, t Y B T Y T X (9) t ( ).( a. t b. b, t ) True values for Y t are generated usng equaton (9) for each perod startng from ntal values of Y from normal random varables, and exogenously generated values for the varable X bt,. Ths t process s terated several tmes to ensure that the pre-determned varable Yt s generated usng the same underlyng spatal process as Y t. The observed value of Y t s subsequently obtaned by

10 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study perturbng ts true values wth dsturbances U ( u, u) whose values were generated n the prevous secton. Y Y U t, observed t t () The results from Monte Carlo smulatons are at best random. In order to obtan suffcently accurate results, a large number of repettons s requred. The errors due to the number of repettons were reduced by use of antthetc varates. Thus, n equaton () both postve and negatve error terms are used to generate the observed values of Y. 3.5 Smulaton desgn Random samples are drawn from a specfed dstrbuton, and a set of data consstent wth the model s generated. It s then used to estmate model parameters usng the FGSSLS and FGS3SLS estmator. Ths process s repeated several tmes. The estmates are then averaged to obtan the expected values of parameters of nterest. The whole process s repeated for varyng degrees of spatal dependences, sample sze and the neghborhood structure to analyze the performance of FGS3SLS estmator under dfferent condtons to analyze the senstvty of the results to the data generaton process. The complete code for the experments s wrtten n the statstcal package R (R Development Core Team, 5). 4. Results Monte Carlo smulaton usng the above parameters and synthetcally generated data s performed for all combnatons of the weght matrx W, sample sze n and the spatal lag parameter. 5 random samples of errors are generated for each set of n,, the neghborhood matrx W and the covarance matrx. Each vector of errors s used twce (as thetc and antthetc varates) resultng n repettons for each experment. Ths setup yelds a total of values for, for W, 9 for repettons for each experment., 9 for and 3 for n resultng n 97 experments wth The performance of the Feasble Generalzed Spatal Three Stage Least Square estmator (FGS3SLS) was found to be superor to the Feasble Generalzed Spatal Two Stage Least Square estmator (FGSSLS) whch n turn was found to be superor to the ordnary two stage

11 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study least square estmator under varyng condtons. Table demonstrates the gans from usng FGS3SLS for the parameter settngs descrbed n ths paper. The estmates from FGS3SLS have smaller bas and varance compared to the FGSSLS estmator. The gans from usng the former are greater when the spatal correlaton n dsturbance terms s hgh, the spatal lag parameter has a low absolute value, the sample sze s small and the neghborhood structure s less dense. <<nsert table here >> Gven the overall superorty of the FGS3SLS estmator under dfferent condtons, we wll only focus on the propertes of FGS3SLS n the subsequent analyss. The smulatons permt analyss of the mpact of sample sze, neghborhood densty, varance-covarance structure of dsturbances and the strength of spatal dependence on parameter estmates obtaned usng ths estmator. 4. Impact of sample sze on parameter estmates In ths secton, we analyze the mpact of sample sze on parameter estmates usng root mean square errors () as a measure of performance for the FGS3SLS estmator. An attempt s made to solate the nteracton effects of sample sze wth neghborhood densty (average number of neghbors), varance covarance matrx of error structures, degree of spatal dependences n endogenous varables and spatal autocorrelaton n errors. For brevty of presentaton, we choose one value of and show the mpact of varyng sample sze on of ˆ for dfferent values of. Smlarly, we choose one value of and show the mpact of varyng sample sze on of ˆ for dfferent values of. The exercse s repeated for the two varance-covarance matrces (see fgure ). <<nsert fgure here>> Increasng the sample sze from to 5 observatons had a huge mpact on the of a parameter estmates rrespectve of other control varables lke neghborhood densty or the varance covarance matrx. However, the gans n ncreasng the sample sze from 5 to 5 were margnal except at extreme values of spatal dependence parameters and. A large sample sze mproves the performance much more when the spatal lag parameter of the

12 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study dependent varable s small, the spatal autocorrelaton n errors s hgh, the number of neghbors s large and the varance covarance structure of error are large. 4. Impact of the average number of neghbors specfed n the weght matrx W The choce of neghborhood structure as defned by W s often decded a pror usng exploratory data analyss or s based on the goodness of ft crtera. Ths s because the data generaton process s not known n practce and the theory behnd selecton of the weght matrx s weak (see Cuaresma, ). Accordng to the smulatons, the mpact of neghborhood densty on of parameter estmates depends on the strength of spatal dependences (, ) as shown n fgure. For all parameter estmates except that of, ncreasng the average number of neghbors ncreased the notceably for the followng two combnatons of spatal dependence parameters (a) extreme negatve values of and hgh postve, and (b) small absolute values of and hgh postve. However, for small absolute values of and extreme negatve values of, the actually decreased. The estmates of condtonal on W behaved slghtly dfferently. Increasng the densty margnally ncreased the for small rrespectve of but was drastcally decreased for extreme negatve values of (except at hgh postve ). << nsert fgure here >> The experments wth dfferent number of average neghbors revealed that as the structure becomes denser, the bas n parameter estmates ncreases many tmes. The effect s more pronounced as the spatal autocorrelaton n the dependent varable and error structure ncreases. An ncrease n the sample sze consstently and greatly reduces the bas due to the ncrease n neghborhood densty. Thus, n a large sample, the ncrease n bas due to a denser neghborhood structure s margnal. The result for estmates of for dfferent values of sample sze and degree of spatal dependences are shown n fgure 3. Estmates of other model parameters behaved n smlar fashon. << nsert fgure 3 here>>

13 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 3 The smulaton results suggest that the choce of neghborhood structure should not only nvolve goodness of ft crtera but also concern for ncreased bas n parameter estmates due to denser neghborhood structure. 4.3 Estmates of and The bas n the estmate of the spatal autocorrelaton parameter n error terms was analyzed under dfferent sample szes, varance-covarance structure and weght matrces condtonal on dfferent values of the spatal lag parameter. Smlar analyss was conducted for the estmates of the spatal lag parameter condtonal on (fgure 4). The estmator does not provde a drect way to calculate the varance of and therefore, t was derved computatonally. One pont of cauton s that the estmaton of requres an optmzaton procedure where the objectve functon may not be well defned and s susceptble to the choce of startng parameters. Ths was not found to be the case n our experments as the results were stable wth respect to the choce of startng parameters. However, t s a concern to be borne n mnd whle usng the estmator. Estmates of were very robust to varyng degrees of spatal dependences over most of the (, ) space. As the neghborhood densty ncreases, there s an ncrease n the bas and s mostly ndependent of the value of on whch t s condtoned. The estmator performs well at low and moderate degrees of spatal dependences n endogenous varables except when there s a smultaneous presence of a very hgh spatal dependence n randomness. Surprsngly, hgher bas n the parameter estmate of was accompaned by hgher varances, sgnfyng the poor performance of the estmator n such condtons. The bas and varance of was largely ndependent of the values of t was condtoned upon except at very hgh values of. The bas ncreased very rapdly when ts true parameter value ncreased from -.8 to +.8. However, unlke, there was a clear bas-varance trade off n the estmates of. <<nsert fgure 4 here>>

14 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 4 5. Concluson In ths paper, we analyzed the small sample propertes of the lmted nformaton Feasble Generalzed Spatal Two Stage Least Squares (FGSSLS) and the full nformaton Feasble Generalzed Spatal Three Stage Least Squares (FGS3SLS) estmator for a system of smultaneous equatons wth spatal dependence n error terms and n the dependent varable. Gven relatvely few publshed applcatons of ths estmator and lack of theoretcal understandng about ts behavor n small samples, ths paper provdes a startng pont for analyzng the behavor of ths estmator. A Monte Carlo framework was used to explore the mpacts of sample sze, neghborhood structure, varance co-varance matrx and varyng degree of spatal dependence parameters on the estmators performance. The FGS3SLS estmator performed better than the FGSSLS estmator n terms of smaller bas and lower varance. The gans of usng the former are hgher when the spatal correlaton n the dsturbance terms s hgh, the spatal lag parameter has a low absolute value, the sample sze s small and the neghborhood structure s dense. Gven the superorty of the FGS3SLS estmator over the FGSSLS n the smulatons descrbed n ths paper, the detaled study of the mpacts of sample sze, neghborhood structure, varance-covarance matrx and degree of spatal dependence on estmator s behavor that was made was lmted to the FGS3SLS estmator. The performance of the FGS3SLS estmator drastcally mproved when the sample sze was ncreased from to 5 observatons. Increasng the sample sze to 5 observatons yelded only margnal gans. Gans wth ncreasng sample sze are more sgnfcant when the heterogenety s hgh, the spatal lag parameter of the dependent varable s small, the spatal autocorrelaton n errors s hgh, the number of neghbors s large and the varance covarance structure of error s large. The performance of the estmator was found to be senstve to the value of the spatal dependence parameters. It deterorated wth low values of the spatal lag parameter n the dependent varable ( ) and at extremely hgh values of the spatal dependence n the error structure ( ). Thus, the estmator pays a premum n terms of bas and varance when the spatal lag s small but has huge gans as the spatal lag ncreases. The estmator for performed well at low and moderate degrees of spatal dependences n the endogenous varables except when there s a smultaneous presence of a very hgh spatal dependence n randomness. Spatal structures wth hgher average number of neghbors led to hgher bas and varances n

15 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 5 the estmates. The effect s more pronounced as the spatal autocorrelaton n the dependent varable and error structure ncreases. In large samples, the ncrease n bas due to denser neghborhood structure s margnal. The results presented here are senstve to the model specfcaton, choce of the data generaton process, dstrbuton of the exogenous varable, etc. However, the results are useful as a comparatve exercse to assess the relatve changes n performance under dfferent condtons and should not be taken as an absolute measure of performance. Understandng the mpacts of the sample sze, varyng degrees of spatal dependences, neghborhood structure and the error structure on the forecasted value s essental. However, the mportance of ths work n analyzng the forecasts of spatal data and comparng wth the results wth true values was not addressed n ths paper. Addtonal research s needed n order to enhance the use of ths estmator n appled work. It s computatonally ntensve and there s no software or standard code to mplement ths estmator. Efforts n ths drecton are very much warranted. A useful extenson would be to analyze the mpact of ncreasng model complexty and choce of nstruments on the performance of the estmator. Further, the estmates of are obtaned from an optmzaton routne, where the objectve functon may have multple optma. In such cases, the parameter estmate of may be susceptble to the choce of startng values and varous technques may be needed to nsure that a global optmum s reached. Ths makes the task more computatonally demandng. Work s also needed to theoretcally corroborate the fndngs of ths paper n a generalzed framework. Over the last fve decades, we have learnt a great deal about the propertes of the three stage least squares estmator n terms of mpacts of msspecfcaton, nonlnearty, multcollnearty, etc., many of whch have been studed through Monte Carlo smulatons. A parallel seres of lterature needs to be developed for the Feasble Generalzed Three Stage Least Square estmator. 6. References Anseln, L. (988) Spatal econometrcs: Methods and models, Boston, Kluwer Academc Publshers. Anseln, L. () Thrty years of spatal econometrcs, Papers n Regonal Scence, 89, 3-5 Carter, R. A. L. & Nagar, A. L. (997) Coeffcents of correlaton for smultaneous equaton systems, Journal of Econometrcs, 6, Case, A. (99) Spatal patterns n household demand, Econometrca, 59, Case, A., Rosen, H. S. & Hnes, J. R. (993) Budget spllovers and fscal polcy ndependence: Evdence from the states, Journal of Publc Economcs, 5, Clff, A. & Ord, J. (973) Spatal autocorrelaton, London, Pon.

16 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 6 Clff, A. & Ord, J. (98) Spatal processes, models and applcatons, London, Pon. Cuaresma, J. C. & Feldkrcher, M. () Spatal flterng, model uncertanty and the speed of ncome convergence n europe. Workng paper 6, Oesterrechsche Natonal Bank Das, D., Kelejan, H. H. & Prucha, I. R. (3) Small sample propertes of estmators of spatal autoregressve models wth autoregressve dsturbances, Papers n Regonal Scence, 8, -6. Davdson, R. & MacKnnon, J. G. (993) Estmaton and nference n econometrcs, New York, Oxford Unversty Press. Drffeld, N. (6) On the search for spllovers from foregn drect nvestment (FDI) wth spatal dependency, Regonal Studes, 4(), 7-9. Fshback, P. V., Horrace, W. C. & Kantor, S. (6) The mpact of New Deal expendtures on moblty durng the great depresson, Exploratons n Economc Hstory, 43(), 79-. Goodchld, M., Anseln, L., Appelbaum, R. & Harthorn, B. () Toward spatally ntegrated socal scence, Internatonal Regonal Scence Revew, 3, Kelejan, H. H. & Prucha, I. R. (999) A generalzed moments estmator for the autoregressve parameter n a spatal model, Internatonal Economc Revew, 4, Kelejan, H. H. & Prucha, I. R. (4) Estmaton of smultaneous systems of spatally nterrelated cross sectonal equatons, Journal of Econometrcs, 8, 7-5. Ngeleza, G. K., Florax, R. J. G. M. & Masters, W. A. (6) Geographc and nsttutonal determnants of real ncome: A spato-temporal smultaneous equaton approach, Workng Papers 6-5, College of Agrculture, Department of Agrcultural Economcs, Purdue Unversty. Phllps, P. C. P. (983) Exact small sample theory n the smultaneous equatons model, n: Z. Grlches & M. Intrlgato (eds.) Handbook of Econometrcs, pp 44-56, Vol, New York, North Holland. R Development Core Team. (5) R: A language and envronment for statstcal computng. R Foundaton for Statstcal Computng, Venna, Austra. ISBN , Avalable from Rothenberg, T. J. (984) Hypothess testng n lnear models when the error covarance matrx s nonscalar, Econometrca, 5, Sarraf, S. () Three essays on socal dynamcs and land-use change: Framework, model and estmator, Unpublshed doctoral dssertaton, Unversty of Illnos, Urbana Champagn.

17 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 7 FGS3SLS Bas Varance Gans over FGSSLS Reducton n Absolute Bas Reducton n Varance Table : FGS3SLS Bas and Varances for n=5, 3 3, W=6,.3,., 3.5

18 Avg. no. of neghbors = 6 Avg. no. of neghbors = Avg. no. of neghbors = Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study n= n=5.5 n= n=5.4 n=5.4 n= lambda lambda = - = n= n=5. n= n=5. n=5. n= Rho Rho = -. = n= n=5.4. n= n=5. n=5. n= Rho Rho = -. = -. Fgure : Impact of sample sze on of

19 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study 9 Average number of neghbors= Average number of neghbors=6 G, FGS3SLS G, FGS3SLS rho rho L, FGS3SLS L, FGS3SLS rho rho rho, FGS3SLS rho, FGS3SLS rho rho Fgure : Impact of average number of neghbors on for and n = 5

20 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study N= Average number of neghbors= G, FGS3SLS Average number of neghbors=6 G, FGS3SLS - - Percent -3 Bas rho Percent Bas rho N= 5 G, FGS3SLS G, FGS3SLS Percent -. Bas rho Percent -.5 Bas rho N= 5 G, FGS3SLS G, FGS3SLS Percent -.4 Bas rho Percent -. Bas rho Fgure 3: Impact of average no. of neghbors on percentage bas of (.7) for

21 Performance of the FGS3SLS Estmator n Small Samples: A Monte Carlo Study Bas Varance L, FGS3SLS. L, FGS3SLS Bas -.8 Varance rho rho r, FGS3SLS. r, FGS3SLS Bas Varance rho rho Fgure 4: Estmates of and, at n=5 for, average number of neghbors = 6