Spatial models with spatially lagged dependent variables and incomplete data

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1 J Geogr Syst (2010) 12: DOI /s ORIGINAL ARTICLE Spatial models wit spatially lagged dependent variables and incomplete data Harry H. Kelejian Ingmar R. Pruca Received: 23 July 2009 / Accepted: 1 February 2010 / Publised online: 6 Marc 2010 Ó Springer-Verlag 2010 Abstract Te purpose of tis paper is to suggest estimators for te parameters of spatial models containing a spatially lagged dependent variable, as well as spatially lagged independent variables, and an incomplete data set. Te specifications allow for nonstationarity, and te disturbance process of te model is specified non-parametrically. We consider various scenarios concerning te pattern of missing data points. One estimator we suggest is based on a smaller but complete subset of te sample anoter is based on a larger but incomplete subset of te sample. We give large sample results for bot of tese cases. Keywords Spatial models Missing data Instrumental variable estimation JEL Classification C21 C31 1 Introduction Missing data problems often arise in te analysis of spatial models containing spatial lags. 1 In some cases, tese data problems arise because of a lack of data on 1 Tere is, of course, a large literature on missing data issues. Some important early studies are Anderson (1957), Friedman (1962), Afifi and Elasoff (1966, 1967, 1969), Haitovsky (1968), and Kelejian (1969). A nice, but sort, review of some of tese early studies is given in Maddala (1977, pp ) Kmenta (1986, pp ) also discusses missing data problems, and igligts certain issues. An excellent recent overview of models and procedures is given in Little and Rubin (2002) anoter interesting text, in a spatial framework, wic primarily focuses on te collection of spatial data is Müller (2007). For a review of studies wic focus primarily on missing data issues in a spatial framework see Anselin (1988, pp ) and te references cited tere-in. A more extensive review of suc studies can be found in Cressie (1993) under te category edge effects in te index of tat book. H. H. Kelejian (&) I. R. Pruca Department of Economics, University of Maryland, College Park, MD 20742, USA kelejian@econ.umd.edu I. R. Pruca pruca@econ.umd.edu

2 242 H. H. Kelejian, I. R. Pruca some variables for units tat are defined to be neigbors of oter units for wic data are available. 2 In some, but not all of tese cases, te missing data relate to edge units see e.g., Anselin (1988, pp ). In oter cases, data sortcomings may arise because of eiter te unavailability of data on certain variables or te reluctance of te researcer to use data on certain variables because of quality of data concerns. One example of tis would be models tat, among oter tings, require data on te GDPs of te sampled countries, and GDP data are eiter not available for some countries or are of dubious quality and so not used see e.g., Kelejian et al. (2008). Tere are various ways researcers ave confronted tis problem. One approac is to ignore it if te missing data relate to units tat are involved as part of a spatial lag. In tis approac, te missing observations are implicitly replaced by zeroes and so, e.g., te spatial lag would be constructed entirely in terms of te available data. Anoter approac is to complete te sample using available data by estimating in various ways, te observations tat are missing. Still anoter approac is to rely on maximum likeliood estimation. 3 Of course, tere are still oter approaces, but formal results concerning te distribution of te regression parameter estimators based on incomplete data sets are not available in te framework of a non-stationary spatial model tat contains spatial lags in bot te dependent and independent variables, as well as a non-parametrically specified disturbance process. 4 Te purpose of tis study is to fill tis gap. For suc non-stationary spatial models, we discuss various scenarios relating to te missing data and te manner in wic te sample increases. We suggest regression parameter estimators tat are based on a complete subset of te sample, as well as on a larger but incomplete subset of te sample. We give formal large sample results for our suggested estimators in bot of tese cases. In te context of our model, we give conditions under wic te effects of missing observations in te estimation procedure are asymptotically negligible, as well as wen tey are not! Our asymptotic results account for bot of tese cases. Finally, we give user friendly suggestions concerning small sample inferences for our suggested estimators. We specify te model in Sect. 2. Tat section also contains a discussion of various scenarios as to ow te sample may increase. Our suggested estimators are given in Sect. 3, along wit teorems tat describe teir large sample distribution. Tese teorems relate to te various scenarios concerning te manner in wic te sample increases. Conclusions are given in Sect. 4. Tecnical details are relegated to te Appendix. 2 As one example, in edonic models of ousing prices involving a spatially lagged dependent variable, te prices of unsold ouses would not be known. In some of tese cases, te problem would be to predict te prices of te unsold omes given teir attributes and te spatial interdependence described by te weigting matrix see e.g., LeSage and Pace (2004). See also Kelejian and Pruca (2007) for issues relating to suc prediction. 3 See e.g., te review in Anselin (1988, pp ), te discussions in Cressie (1993) under te category edge effects in te index of tat book, and te various procedures described by Haining (2003). 4 We note tat te specification of te regression model is typically based on economic teoretical reasoning te disturbance process is te residual in te model, and typically teoretical reasoning does not apply to it. Hence, a non-parametrically specified disturbance process sould, peraps, be considered more often tan it is!

3 Spatial models wit spatially lagged dependent variables and incomplete data Model 2.1 Specification In tis section, we specify a model containing spatial lags in te dependent variable, as well as in some of te independent variables. We describe tis model in suc a way tat for certain units, te dependent variable and all of te regressors determining tat dependent variable, including te spatially lagged variables, are observed. For oter units, observations on eiter te dependent variable and/or all of te regressors determining tose dependent variables are not available. More specifically, we divide te units into tree groups. For units in groups 1 and 2, te data on te dependent and exogenous variables pertaining to tose units are observed. Te distinction between units in group 1 and 2 is tat for te former all observations needed to formulate spatial lags are also observed, wile for te latter, tis is not te case. Te observations pertaining to units in group 3 are unobserved. As in illustration of suc a grouping of te data, consider Fig. 1. If we assume tat spatial interactions can be described by a rook design, ten in tis illustration, units in group 1 and group 3 are not immediate neigbors, and tus te corresponding block in te spatial weigts matrix will be zero. Consequently, we can compute spatial lags of te dependent and exogenous variables for all units in group 1, despite tat all variables pertaining to group 3 are unobserved. Tis is not te case for units in group 2, since tose units are immediate neigbors in bot groups 2 and 3, and tus spatial lags would depend on observations from units in group 3. Consistent wit te above illustrative example, we assume in te following discussion tat te data are ordered suc tat te (1, 3)-block in te spatial weigts Fig. 1 An exemplary grouping of data on a rectangular grid wit neigbors defined by a rook design

4 244 H. H. Kelejian, I. R. Pruca matrix corresponding to groups 1 and 3 is zero. Since it as no effect on te derivations relating to our large sample results, we allow for te (3, 1)-block in te spatial weigts matrix to be non-zero for purposes of generality, altoug we expect it to be zero in most applications. In particular, we consider te following model: y n1 X n1 W n11 W n12 0 J n1 B C B C B CB y n2 A X n2 Ab 1 W n21 W n22 W n23 A@ J n2 Ab 2 y n3 X n3 W n31 W n32 W n W n11 W n12 0 y n1 B CB C B þ k@ W n21 W n22 W n23 A@ y n2 A J n3 u n1 u n2 1 C A ð1þ W n31 W n32 W n33 y n3 u n3 and u n1 R n11 R n12 R u n2 A R n21 R n22 R n23 A e 1 e n2 A u n3 R n31 R n32 R n33 e n3 were y n,i and X n,i are, respectively, te n i 9 1 vectors of endogenous variables and n i 9 s matrices of exogenous variables corresponding to group i for i = 1, 2, 3. Te researcer observes y n,1, y n,2, X n,1 and X n,2, but not y n,3, and X n,3. Te matrices W n,ij i, j = 1, 2, 3 are observed nonstocastic weigting matrices, were W n,13 = 0. Te matrices J n,i are of dimension n i 9 r wit r B s and represent submatrices of X n,i. Hence, te above specification allows for spatial lags in some, but not necessary all, exogenous regressors. Of course, given tat te matrices J n,i are submatrices of X n,i, it follows tat J n,1 and J n,2 are observed, but not J n,3. We conditionalize on te exogenous variables and so take te matrices X n,i and J n,i, i = 1, 2, 3 as nonstocastic. For future reference, let n = n 1? n 2? n 3, and let W n and R n be te n 9 n matrices W n11 W n12 0 R n11 R n12 R n13 W n W n21 W n22 W n23 A R n R n21 R n22 R n23 A ð2þ W n31 W n32 W n33 R n31 R n32 R n33 Given te notation in (2), te model in (1) can be written more compactly as y n ¼ X n b 1 þ W n J n b 2 þ kw n y n þ u n ¼ Z n c þ u n ð3þ u n ¼ R n e n were y 0 n ¼ y0 n1 y0 n2 y0 n3 Xn 0 ¼ X0 n1 X0 n2 X0 n3 Jn 0 ¼ J0 n1 J0 n2 J0 n3 Z n ¼ðX n W n J n W n y n Þ c 0 ¼ b 0 1 b0 2 k u 0 n ¼ðu0 n1 u0 n2 u0 n3 Þ e0 n ¼ e0 n1 e0 n2 e0 n3 :

5 Spatial models wit spatially lagged dependent variables and incomplete data 245 We will suggest two instrumental variable estimators for situations were some of te data are missing. Tese estimators presume tat te researcer is able to arrange te data as specified in (1). One of tem will be based on te complete portion of te sample, i.e., te portion of te sample for wic te dependent variable, te exogenous variables and te spatial lags are observed. Te dependent variable corresponding to tis estimator is te n vector y n,1. Te oter estimator will be based on a larger but incomplete portion of te sample, were te dependent and exogenous variables are observed, but were some of te data needed to compute te spatial lags on te r..s. are unobserved. Te dependent variable corresponding to tis estimator is te n 1? n vector ½y 0 n1 y0 n2 Š. For notational convenience, we define i 0 y n1þ2 ¼ y 0 n1 y0 n2 ð4þ and correspondingly i 0 X n1þ2 ¼ Xn1 0 X0 n2 i 0 J n1þ2 ¼ Jn1 0 J0 n2 ð5þ i 0: u n1þ2 ¼ u 0 n1 u0 n2 We also define W n,1 as tat portion of te weigting matrix, wic determines te spatial lags in te sub-model for y n,1 as implied by (1), namely W n1 ¼ W n11 W n12 : ð6þ In a similar way, W n,1?2 is defined as tat portion of te weigting matrix, wic determines te spatial lags in te sub-model for y n,1?2 as implied by (1), namely W n1þ2 ¼ W n11 W n12 0 : ð7þ W n21 W n22 W n23 Tis notation is extended to submatrices of R n as defined in (2) e.g., R n1 ¼ ðr n11 R n12 R n13 Þ ð8þ R n1þ2 ¼ R n11 R n12 R n13 : R n21 R n22 R n23 In terms of tis notation, and borrowing oter notation from Kelejian and Pruca (1998) by letting J n1 ¼ W n1 J n1þ2 and y n1 ¼ W n1 y n1þ2 te model in (1) implies tat y n,1 is determined as y n1 ¼ X n1 b 1 þ J n1 b 2 þ ky n1 þ u n1 Z n1 c þ u n1 ð9þ

6 246 H. H. Kelejian, I. R. Pruca were Z n1 ¼ X n1 J n1 y n1 c 0 ¼ b 0 1 b0 2 k : Te model in (1) implies furtermore tat y n,1?2 is determined as y n1þ2 ¼ X n1þ2 b 1 þ W n1þ2 J n b 2 þ kw n1þ2 y n þ u n1þ2 : ð10þ Assuming W n,23 = 0, te regressors in te (10) are not all observed. Part of our large sample results reported below relate to conditions under wic te unobserved parts of (10) are asymptotically negligible. Tat analysis is facilitated by separating te observed portions of te regressors in (10) from te unobserved portions. Hence, o u we define W n,1?2 and W n,1?2 as, respectively, tose parts of W n,1?2 tat relate to te observable and unobservable regressors, namely Wn1þ2 o ¼ W n11 W n12 W n21 W n22 Wn1þ2 u ¼ 0 W n23 Given tis notation, and again borrowing notation from Kelejian and Pruca (1998) by letting J n1þ2 ¼ Wn1þ2 o J n1þ2 and y n1þ2 ¼ Wn1þ2 o y n1þ2 te model in (10) can be rewritten as y n1þ2 ¼ X n1þ2 b 1 þ J n1þ2 b 2 þ ky n1þ2 þ n n ¼ Z n1þ2 c þ n n ð11þ were Z n1þ2 ¼ X n1þ2 J n1þ2 y n1þ2 i n n ¼ Wn1þ2 u J n3b 2 þ kwn1þ2 u y n3 þ u n1þ2 and were n n in (11) can be viewed as a contaminated error vector. Bot (9) and (11) contain spatial lags in te dependent variable, and so our suggested estimators will involve instrument matrices. Consistent wit our earlier notation, te instrument matrices for te estimation of (9) and (11) are, respectively, te n 1 9 q 1 and n 1? n 2 9 q 2 matrices H n,1 and H n,1?2. Ideally, we would like to use te conditional means of y n1 and y n1þ2 as instruments. Tese conditional means depend on X 3,n and are ence unobserved. Guided in part by suggestions in Kelejian and Pruca (1998), our recommendation ere is to take as instruments te linearly independent columns of H n1 ¼ X n1 W n11 X n1...wn11 s X n1 J n1 W n11 J n1...wn11 s J i n1 i ð12þ H n1þ2 ¼ X n1þ2 Wn1þ2 o X n1þ2...ðwn1þ2 o Þs X n1þ2 were typically s B 2. Note tat since (9) contains te n endogenous regressor y n1 ¼½W n11 y n1 þ W n12 y n2 Š, one migt also add W n,12 X n,2 to te columns of te instrument matrix H n,1 to elp account for te mean of W n,12 y n,2. Because W n,12 is an n 1 9 n 2 matrix and X n,1 is an n 1 9 s matrix, W n,12 is not conformable for

7 Spatial models wit spatially lagged dependent variables and incomplete data 247 postmultiplication by X n,1. Typically, n 2 will be less tan n 1 we do not suggest constructing furter instruments by postmultiplying augmented versions of W n,12 by X n, Te sample configurations Our large sample results below relate to te relative sample sizes n i, i = 1, 2, 3. We consider tree cases. Specifically, we assume te configuration of te space containing te n units expands in suc a way tat Case 1: n 1?? n Case 2: n 1!1 and 3! 0 ðn 1 þn n 2 Þ 1=2 Case 3: n 1!1 and 3! c [ 0: ðn 1 þn 2 Þ 1=2 Case 1 relates to te use of y n,1 as te dependent variable. In tis case, tere is no assumption concerning te magnitudes of n 2 or n 3 and, ence, tey bot could increase of (any order) beyond limit! As a preview, te reason tese magnitudes do not matter relates to our assumptions below concerning te weigting matrix, W n. Cases 2 and 3 relate to te use of y n,1?2 as te dependent variable. Note tat case 2 does not rule out n 2??, n 3??, or even n i /n 1??, i = 2, 3. Case 3 differs from case 2 in tat te missing portion of te sample, referenced ere by n 3, increases in te same order as te square root of te observed portion of te sample indexed ere by n 1? n Assumptions In tis section, we state our assumptions teir interpretations are given in te following section. Let D n1 ¼ X n1 J n1 and D n1þ2 ¼ X n1þ2 J n1þ2 : Our assumptions are given below. Assumption 1 Te elements of e n, namely {e i,n :1 B i B n, n C 1}, are identically distributed. Furter, {e i,n :1 B i B n} are for eac n distributed (jointly) independently wit E(e i,n ) = 0, E(e 2 i,n ) = 1. Additionally, te innovations are assumed to possess a finite fourt moment. Assumption 2 Te elements of X n, H n,1, and H n,1?2 are uniformly bound in absolute value in addition, D n,1 and D n,1?2 ave full column rank for n 1 and n 2 large enoug. In te limits below, no restrictions are placed on te magnitudes of n 2 and n 3, wic may be related to n 1 and so may also increase beyond limit. Assumption 3 We assume (a) lim n1!1 n 1 1 D0 n1 D n1 ¼ Q D1 D 1 (b) lim n1!1 n 1 1 H0 n1 H n1 ¼ Q H1 H 1 (c) lim n1!1ðn 1 þ n 2 Þ 1 D 0 n1þ2 D n1þ2 ¼ Q D1þ2 D 1þ2 (d) lim n1!1ðn 1 þ n 2 Þ 1 Hn1þ2 0 H n1þ2 ¼ Q H1þ2 H 1þ2

8 248 H. H. Kelejian, I. R. Pruca were Q D1 D 1 Q D1þ2 D 1þ2 Q H1 H 1 and Q H1þ2 H 1þ2 are finite nonsingular matrices. Assumption 4 (a) k \ 1 (b) Te diagonal elements of W n are zero, and R n and (I n - kw n ) are nonsingular (c). Te row and column sums of R n, W n and (I n - k W n ) -1 are uniformly bound in absolute value. Assumption 5 We assume (a) p lim n1!1 n 1 1 H0 n1 Z n1 ¼ Q H1 Z 1 (b) lim n1!1 n 1 1 H0 n1 R n1r 0 n1 H n1 ¼ Q H1 R 1 R 1 H 1 (c) p lim n1!1ðn 1 þ n 2 Þ 1 Hn1þ2 0 Z n1þ2 ¼ Q H1þ2 Z 1þ2 (d) lim n1!1ðn 1 þ n 2 Þ 1 Hn1þ2 0 R n1þ2r 0 n1þ2 H n1þ2 ¼ Q H1þ2 R 1þ2 R 1þ2 H 1þ2 were Q H1 Z 1 Q H1 R 1 R 1 H 1 Q H1þ2 Z 1þ2 and Q H1þ2 R 1þ2 R 1þ2 H 1þ2 are finite matrices wit full column rank. Te next assumption is only maintained under Case 3. Assumption 6 n For n 1?? and 3! c wit 0 \ c \?, we assume ðn 1 þn 2 Þ 1=2 (a) lim n1!1ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 J n3 ¼ F 1þ2 (b) lim n1!1ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 ði n kw n Þ 1 n3 X n ¼ L 1þ2 (c) lim n1!1ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 ði n kw n Þ 1 n3 W nj n ¼ S 1þ2 were F 1?2, L 1?2, and S 1?2 are, respectively, q 2 9 r, q 2 9 s, and q 2 9 r finite matrices, and were (I n - kw n ) -1 n,3 is te n 3 9 n matrix consisting of te last n 3 rows of (I n - kw n ) A brief discussion of te assumptions First note from (3), Assumptions 1 and part (b) of Assumption 4 tat Ey n ¼ðI n kw n Þ 1 ½X n b 1 þ W n J n b 2 Š ð13þ Assumption 2 and part (c) of Assumption 4 imply tat te elements of Ey n are uniformly bound in absolute value. Also note tat Assumption 1 and part (c) of Assumption 4 imply tat te row and column sums of te variance covariance (encefort, VC) matrix of y n, say VC y_n, are uniformly bound in absolute value, were VC yn ¼ ði n kw n Þ 1 R n R 0 n I n kwn 0 1: ð14þ Assumptions 1, 2, 3(a), and 4 are standard-type conditions tat ave been discussed elsewere in te literature, see e.g., Kelejian and Pruca (1998, 2004). Note also tat te condition E(e 2 i,n ) = 1 in Assumption 1 is not restrictive, since te only assumption maintained for R n is tat its row and column sums are uniformly bounded. Tus, te variance of te e i,n is a scale factor, wic can be incorporated into R n. To see tis, note tat if E(e 2 i,n ) = r 2 e, te disturbance vector u n in (3) could be redefined as:

9 Spatial models wit spatially lagged dependent variables and incomplete data u n ¼ R n e n ¼½r e R n Š e n ¼ R n r e n e were te row and column sums of R * n are again uniformly bounded, and te elements of e * n would ave a variance of 1.0. Consider part (c) of Assumption 3. Te assumption requires tat ðn 1 þ n 2 Þ 1 Xn1þ2 0 X n1þ2 ðn 1 þ n 2 Þ 1 Xn1þ2 0 W n1þ2 o X n1þ2 and ðn 1 þ n 2 Þ 1 Xn1þ2 0 W n1þ2 o0 Wo n1þ2 X n1þ2 converge. Tese limit assumptions are standard-type conditions. Given part (a) of Assumption 3, and if and if n 2 /n 1?0, ten Q D1þ2 D 1þ2 ¼ Q D1 D 1. Similar comments apply to parts (b) and (d) of Assumption 3. Now consider part (a) of Assumption 5, and note tat n 1 1 H0 n1 Z n1 ¼ n 1 1 H0 n1 ðx n1 W n1 J n1þ2 W n1 y n1þ2 Þ: ð15þ Since assumptions concerning te limit of terms suc as n 1 1 H0 n1 X n1 are standard, we will focus our attention on te oter two components in (15), namely w n1 ¼ n 1 1 H0 n1 W n1j n1þ2 and / n1 ¼ n 1 1 H0 n1 W n1y n1þ2 : Recalling tat te columns of J n,1?2 are a subset of tose in X n,1?2, it follows from Assumption 2 and part (c) of Assumption 4 tat, regardless of te relative magnitudes of n 2 and n 1, te elements of w n,1 are uniformly bound in absolute value. Hence, at tis point, part (a) of Assumption 5 imposes te condition tat te limits of te elements of w n,1 exist. Now, consider / n,1 and note Eð/ n1 Þ¼n 1 1 H0 n1 W n1e½y n1þ2 Š ð16þ were E[y n,1?2 ] represents te first n 1? n 2 elements of Ey n, wic is given in (13). Again, Assumption 2 and part (c) of Assumption 4 imply tat te elements of E[y n,1?2 ] are uniformly bound in absolute value. Now note, using evident notation, tat VC /n1 i ¼ n 2 1 H0 n1 W n1vc yn1þ2 Wn1 0 H n1 ð17þ were VC yn1þ2 is te upper n 1? n 2 9 n 1? n 2 block of VC yn, wic is given in (14). Assumption 2 and part (c) of Assumption 4 imply tat te elements of VC /n1 are O(n -1 ) and terefore VC /n1? 0, as n 1??. Hence, part (a) of Assumption 5 imposes te condition tat te limits of te elements of E/ n,1 exist. Terefore, in essence, part (a) of Assumption 5 requires tat E(/ n,1 ) converge to a limit, wic is linearly independent of te limit of n 1 1 H0 n1 X n1. Similar arguments suggest tat te oter parts of Assumption 5 are reasonable. Finally, consider Assumption 6, and recall tat tis assumption relates to te case in n wic n 1?? and 3! c, were c is a finite non-zero constant. In reference to ðn 1 þn 2 Þ 1=2 part (a) of Assumption 6, note tat Assumption 2 and part (c) of Assumption 4 imply tat te elements of te q 2 9 r matrix Hn1þ2 0 Wu n1þ2 J n3 are O(n 3 ) or equivalently O((n 1? n 2 ) 1/2 ). Terefore, te elements of ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 J n3 are O(1). Similarly, in parts (b) and (c) of Assumption 6, te elements of te q 2 9 s and q 2 9 r matrices, namely ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 ði n kw n Þ 1 n3 X n and

10 250 H. H. Kelejian, I. R. Pruca ðn 1 þ n 2 Þ 1=2 H 0 n1þ2 Wu n1þ2 ði n kw n Þ 1 n3 W nj n respectively, are O(1). Given tese results, te assumed limits in Assumption 6 are reasonable. 3 Suggested estimators In tis section, we suggest two estimators. One is based on te complete subset of te sample in wic te dependent variable is y n,1. In tis case, te sample size is n 1, and all observations on te regressors determining y n,1 are available. Te oter estimator is based on a larger subset of te sample, wic involves missing data. In tis case, te sample size is n 1? n 2, and te dependent variable is y n,1? Te estimator based on y n,1 1Hn1 Consider te model in (9), and let P nh1 ¼ H n1 Hn1 0 H n1 and ^Z n1 ¼ P nh1 Z n1 : Our suggested estimator of c, wic is based on te complete portion of te sample is ^c 1 ¼ ^Z n1 0 ^Z 1 n1 ^Z n1 0 y n1 ð18þ Te proof of Teorem 1 is given in te Appendix. Teorem 1 Given (9), and Assumptions 1, 2, 3(a), (b), 4, and 5(a), (b) n 1=2 1 ð^c 1 cþ! D N 0 ^Z 1 ^Z 1 X 1 ^Z 1 ^Z 1 ð19þ as n 1?? were and Q ^Z 1 ^Z 1 ¼ Q 0 H 1 Z 1 H 1 H 1 Q H1 Z 1 X 1 ¼ Q 0 H 1 Z 1 H 1 H 1 Q H1 R 1 R 1 H 1 H 1 H 1 Q H1 Z 1: Small sample inferences could be based on ^c 1 N c ^ VC^c1 were wit ^ VC^c1 ¼ n 1 G 0 n1 ^Q H1 R 1 R 1 H 1 G n1 ð20þ 1H G n1 ¼ Hn1 0 H 0 n1 n1 Z n1 ^Z n1 0 ^Z 1 n1 and were ^Q H1 R 1 H 1 R 1 is a consistent estimator of Q H1 R 1 H 1 R 1. Note, given furter assumptions especially relating to distance measures, Q H1 R 1 H 1 R 1 can be estimated

11 Spatial models wit spatially lagged dependent variables and incomplete data 251 non-parametrically as a SHAC estimator as considered in Kelejian and Pruca (2007), because R n1 R 0 n1 is te n 1 9 n 1 VC matrix of u n,1, and u n,1 can be estimated via (9). 5 Of course, in te special case were te disturbances are not spatially correlated and tus Q H1 R 1 R 1 H 1 ¼ r 2 Q H1 H 1, we can employ te estimator ^Q H1 R 1 R 1 H 1 ¼ n 1 1 bu0 n1 bu n1 n 1 Hn1 0 H n1 were bu n1 denotes te estimated disturbances via (9). Remark 1 Note tat te result in (20) relates to te model in (9). Te dependent variable in tat model is y n,1, wic is an n vector on te oter and, te endogenous spatial lag in tat model, namely y n1 ¼ W n1 y n1þ2 involves te (n 1? n 2 ) 9 1 vector y n,1?2. Te result in (20) olds for all possible values of n 2 including n 2 /n 1?? - i.e., te incomplete portion of te sample can be infinitely large relative to te complete portion. A crucial assumption underlying tis result is part (c) of Assumption 4, namely tat te row and column sums of te weigting matrix W n are uniformly bound in absolute value. 3.2 Te estimator based on y n,(1,2) and n 3 ðn 1 þn 2 Þ 1=2!c If n 2 is large, a researcer migt be tempted to estimate te parameters of te model in terms of a sample tat is larger tan just te first n 1 observations. Given te structure of te model, in suc a case, te researcer migt estimate te parameters of te model in terms of (11) instead of (9). In tis case, owever, it is clear from (11) tat missing data problems arise if W n,23 = 0. In tis section, we give results tat correspond to two cases. In te first case, te missing data are of no consequence in te sense tat tey are asymptotically negligible. In tis case, c = 0. In te second case, te missing data are of consequence. In tis case, c is a finite non-zero constant Te case c = 0 1H Consider te model in (11), and let P nh1þ2 ¼ H n1þ2 Hn1þ2 0 H 0 n1þ2 n1þ2 and ^Z n1þ2 ¼ P nh1þ2 Z n1þ2 : Our suggested estimator of c is te two-stage least squares estimator ^c 2 ¼ ^Z n1þ2 0 ^Z 1 n1þ2 ^Z n1þ2 0 y n1þ2 ð21þ Te proof of Teorem 2 is given in te Appendix. Teorem 2 Given Assumptions 1, 2, 3(c), (d), 4 and 5(c), (d) ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ! D N 0 ^Z 1þ2 ^Z 1þ2 X 2 ^Z 1þ2 ^Z 1þ2 ð22þ 5 Te proof of te consistency of te SHAC estimator in Kelejian and Pruca (2007) is based on te assumption tat te disturbance vector and te innovation vector are of te same dimension. Te VC matrix appearing in (22) relates to u n,1 wic is n However, from (1) it is clear tat te innovation vector defining u n,1 as dimension n 1? n 2? n Tis difference requires tedious, but straigt forward, adjustments of te formal proof of consistency of te SHAC estimator given in Kelejian and Pruca (2007). Essentially, te reason for tis is part (c) of Assumption 4.

12 252 H. H. Kelejian, I. R. Pruca as n 1?? were X 2 ¼ Q 0 H 1þ2 Z 1þ2 Q H1þ2 R 1þ2 R 1þ2 H 1þ2 Q H1þ2 Z 1þ2 and Q ^Z 1þ2 ^Z 1þ2 i ¼ Q 0 H 1þ2 Z 1þ2 Q H1þ2 Z 1þ2 Small sample inference can be based on ^c 2 N c ^ VC^c2 ð23þ were wit ^V^c2 ¼ðn 1 þ n 2 ÞG 0 n1þ2 ^Q H1þ2 R 1þ2 R 1þ2 H 1þ2 G n1þ2 ð24þ G n1þ2 ¼ðHn1þ2 0 H n1þ2þ 1 Hn1þ2 0 Z n1þ2 ^Z n1þ2 0 ^Z n1þ2 1 and were ^Q H1þ2 R 1þ2 R 1þ2 H 1þ2 is a consistent estimator of Q H1?2 R 1?2 R 1?2 H 1?2. Remark 2 Observe tat te difference between te disturbance vector and te residual vector in (11) is given by n n u n1þ2 ¼ Wn1þ2 u J n3b 2 þ kwn1þ2 u y n3: ð25þ We conjecture tat given additional assumptions, we sould again be able to estimate te matrix Q H1þ2 R 1þ2 R 1þ2 H 1þ2 non-paramterically via te SHAC estimation approac put forward in Kelejian and Pruca (2007) Te case c = 0 In tis case, te large sample distribution ^c 2 is te same as in (22) except tat te mean of tat large sample distribution is not zero. In particular, te proof of Teorem 3 is in te Appendix. n Teorem 3 Given tat 3! c 6¼ 0 and Assumptions 1, 2, 3(c), (d), 4, ðn 1 þn 2 Þ 1=2 5(c), (d), and Assumption 6 ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ! D N l ^Z 1þ2 ^Z 1þ2 X 2 ^Z 1þ2 ^Z 1þ2 l ¼ ^Z 1þ2 ^Z 1þ2 Q 0 Z 1þ2 H 1þ2 F 1þ2 b 2 ð26þ i þkq 0 Z 1þ2 H 1þ2 ðl 1þ2 b 1 þ S 1þ2 b 2 Þ Remark 3 Note tat (26) implies tat ^c 2 is consistent. However, te mean of te large sample distribution involves limits tat involve X n,3, wic, if X n,3 is not observed, suggests tat te result in (26) is of limited use for making inferences

13 Spatial models wit spatially lagged dependent variables and incomplete data 253 concerning c. Of course, if (n 1? n 2 ) is large, small sample inferences can be l based on (23), wic would be te approximation taken if n 1 þn Summary and suggestions for furter researc In tis paper, we suggest estimators for models tat ave a spatial lag in te dependent variable, a non-parametrically specified disturbance term, and missing observations. Te specification of te disturbance term is suc tat it allows for bot spatial correlation and eteroskedasticity. We consider various configurations of te missing observations. In one case, te missing observations are asymptotically negligible in anoter case, tey are not asymptotically negligible. One of our suggested estimators is based on a small but complete portion of te sample our oter suggested estimator is based on a larger sample tat includes bot te complete portion of te sample, as well as part of te incomplete portion. We give formal large sample results for our suggested estimators and suggest small distribution approximations tat can be used for purposes of inference. Our large sample results account for te various configurations of te missing observations. Among oter tings, a Monte Carlo study tat focuses on te small sample properties of our suggested estimators would be of interest. Our results suggest tat suc a study sould consider various configurations of te missing observations, as well as various specifications of te disturbance term. Specifically, small sample results relating to our estimators for cases in wic te complete portion of te sample is large, but yet small relative to te incomplete portion sould be of particular interest. It sould also be of interest to consider various specifications of te disturbance term involving spatial correlation, etc. and corresponding ypotesis tests relating to te regression parameters based on our small sample approximations. Acknowledgments We gratefully acknowledge financial support from te National Institute of Healt troug te SBIR grants R43 AG and R44 AG We would like to tank two referees for insigtful comments on an earlier draft. Appendix Proof of Teorem 1 It follows from (18) and from (9) tat n 1=2 1 ð^c 1 cþ ¼n 1 ^Z n1 0 ^Z 1n 1=2 n1 ^Z n1 0 u n1 ¼ n 1 1 ^Z n1 0 ^Z 1n 1n ð27þ 1 n1 Zn1 0 H n1 n 1 1 H0 n1 H n1 1=2 Hn1 0 R n1e n Observe tat n 1 1 ^Z n1 0 ^Z i i 1 i n1 ¼ n 1 1 Z0 n1 H n1 n 1 1 H0 n1 H n1 n 1 1 H0 n1 Z n1 : Assumptions 3(a), (b) and 5(a) imply

14 254 H. H. Kelejian, I. R. Pruca n 1 1 ^Z n1 0 ^Z n1! P Q 0 H 1 Z 1 H 1 H 1 Q H1 Z 1 Q ^Z 1 ^Z 1 n 1 1 H0 n1 Z n1! P Q H1 Z 1 n 1 1 H0 n1 H n1! Q H1 H 1 ð28þ were Q ^Z 1 ^Z 1 is a positive definite matrix. Assumptions 2 and 4(c) imply tat te elements of Hn1 0 R n1 are uniformly bound. Given tis, and Assumptions 1 and 5(b), and te central limit teorem in Kelejian and Pruca (1998), it follows tat n 1=2 H 0 n1 R n1e n! D Nð0 Q H1 R 1 R 1 H 1 Þ ð29þ Te proof of Teorem 1 follows from (27) (29). Proof of Teorem 2 Recall tat under Teorem 2, we ave c = 0. It follows from (11) and (21) tat ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ ¼ ðn 1 þ n 2 Þ ^Z n1þ2 0 ^Z 1 i n1þ2 ðn 1 þ n 2 Þ 1=2 ^Z n1þ2 0 n n ð30þ u u were it sould be recalled tat n n = [W n,1?2 J n,3 b 2? k W n,1?2 y n,3? u n,1?2 ] and u n,1?2 = R n,1?2 e n. Consider te first bracketed term in (30). Assumptions 3(d) and 5(c) imply tat ðn 1 þ n 2 Þ 1 ^Z n1þ2 0 ^Z n1þ2! P Q 0 H 1þ2 Z 1þ2 H 1þ2 H 1þ2 Q H1þ2 Z 1þ2 Q ^Z 1þ2 ^Z 1þ2 ðn 1 þ n 2 Þ 1 Hn1þ2 0 Z n1þ2! P Q H1þ2 Z 1þ2 ðn 1 þ n 2 Þ 1 Hn1þ2 0 H n1þ2! Q H1þ2 H 1þ2 ð31þ were Q ^Z 1þ2 ^Z 1þ2 is a finite and nonsingular matrix. Now, consider te second bracketed term in (30). In ligt of (11), i ðn 1 þ n 2 Þ 1=2 ^Z n1þ2 0 n n ¼ðn 1 þ n 2 Þ 1=2 ^Z n1þ2 0 Wu n1þ2 J n3b 2 þ kwn1þ2 u y n3 þ u n1þ2 ¼ðn 1 þ n 2 Þ 1=2 ^Z n1þ2 0 u n1þ2 þ D n i D n ¼ðn 1 þ n 2 Þ 1=2 ^Z n1þ2 0 Wu n1þ2 J n3b 2 þ kwn1þ2 u y n3 : ð32þ We sow below tat te term D n in (32) limits to zero in probability so tat ðn 1 þ n 2 Þ 1=2 ^Z 0 n1þ2 n n ðn 1 þ n 2 Þ 1=2 ^Z 0 n1þ2 u n1þ2! P 0 ð33þ and so, from (30) to(33) ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ ^Z 1þ2 ^Z 1þ2 ðn 1 þ n 2 Þ 1=2 ^Z 0 n1þ2 u n1þ2! P 0: ð34þ Consequently, te two terms on te l..s. of (34) ave te same limiting distribution. In determining tis distribution, note from Assumptions 3(d) and 5(c) tat

15 Spatial models wit spatially lagged dependent variables and incomplete data 255 ðn 1 þ n 2 Þ 1=2 ^Z 0 n1þ2 u n1þ2 Q 0 H 1þ2Z 1þ2 H 1þ2H 1þ2 ðn 1 þ n 2 Þ 1=2 H 0 n1þ2 R n1þ2e n! P 0 provided te last term on te r..s. converges in distribution. To see tis, observe tat Assumptions 2 and 4(c) imply tat te elements of te q 2 9 n 1? n 2? n 3 matrix H 0 n1þ2 R n1þ2 are uniformly bound. Terefore, Assumptions 1, 5(d), and te central limit teorem in Kelejian and Pruca (1998) imply tat ð35þ ðn 1 þ n 2 Þ 1=2 H 0 n1þ2 R n1þ2e n! D Nð0 Q H1þ2 R 1þ2 R 1þ2 H 1þ2 Þ ð36þ Teorem 2 follows from (34) to(36). Proof tat D n! P 0: We now sow tat indeed D n defined in (32) converges in probability to zero as claimed. Recalling tat ^Z n1þ2 ¼ P nh1þ2 Z n1þ2 Assumptions 3(d) and 5(c) imply D n D n!p 0 D n ¼ D n1 þ D n2 D n1 ¼ Q0 Z 1þ2 H 1þ2 ðn 1 þ n 2 Þ 1=2 H 0 n1þ2 Wu n1þ2 J n3b 2 D n2 ¼ kq0 Z 1þ2 H 1þ2 ðn 1 þ n 2 Þ 1=2 H 0 n1þ2 W u n1þ2 y n3 ð37þ Recalling tat te columns of J n are a subset of tose in X n, Assumptions 2 and 4(c) imply tat te elements of q 2 9 r nonstocastic matrix, Hn1þ2 0 Wu n1þ2 J n3 are O(n 3 ). n Terefore, if 3! 0, it follows tat ðn 1 þn 2 Þ 1=2 ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 J n3! 0 ð38þ and so D n1! 0: ð39þ Now consider D n2 : From (13), ED n2 ¼ kq0 Z 1þ2 H 1þ2 ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 Eðy n3þ Eðy n3 Þ¼ðI n kw n Þ 1 n3 ½X ð40þ nb 1 þ W n J n b 2 Š were it sould be recalled tat (I n - kw n ) -1 n,3 is te last n 3 rows of (I n - kw n ) -1, and E(y n,3 )isann vector. Assumptions 2 and 4(c) imply tat te elements of E(y n,3 ) are uniformly bound in absolute value terefore, te elements of te q vector ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 Eðy n n3þ are Oð 3 Þ and ence, ðn 1 þn 2 Þ 1=2 lim n 1!1 ED n2 ¼ 0 ð41þ n if 3! 0. Now consider te variance covariance matrix of D ðn 1 þn 2 Þ 1=2 n2 say VC D.In n2 (37), let Mn 0 ¼ Q0 Z 1þ2 H 1þ2 Hn1þ2 0 Wu n1þ2, and note tat Assumptions 2 and part (c) of 4 imply tat te elements of te s? r? 1 9 n 3 matrix Mn 0 are uniformly bound in absolute value. It follows from (37) tat VC D n2 ¼ k 2 ðn 1 þ n 2 Þ 1 M 0 n VC y n3 M n ð42þ

16 256 H. H. Kelejian, I. R. Pruca were VC yn3 is te lower n 3 9 n 3 block of VC yn given in (15). Assumption 4(c) implies tat te row and column sums of VC yn3 are uniformly bound in absolute value. Terefore, te elements of Mn 0 VC y n3 M n are O(n 3 ) and so VC D n2! 0 ð43þ if n 3! 0: Te results in (41) and (43) imply, via Cebysev s inequality, tat ðn 1 þn 2 Þ 1=2 D n2!p 0 ð44þ It follows from (37), (39), and (44) tat D n! P 0: Proof of Teorem 3 Recall tat under Teorem 3, we ave c [ 0. In ligt of (30) (32), it sould be clear tat te large sample distribution of ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ differs wen c = 0 wen compared to wen c = 0 only because if c = 0 te probability limit of D n is not zero. In tis case, te large sample distribution of ðn 1 þ n 2 Þ 1=2 ð^c 2 cþ as a nonzero mean, wic is te probability limit of D ^Z 1þ2 ^Z n but is oterwise te same as 1þ2 in te case in wic c = 0. To determine tis probability limit, first note tat te limits in (37) do not depend n upon te assumption tat 3! 0. Hence, we determine te probability limit of ðn 1 þn 2 Þ 1=2 D n in terms of tose of D n1 and D n2 : Consider D n1, and note by Assumption 6(a) tat ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 J n3b 2! F 1þ2 b 2. Terefore, in ligt of (37), D n1! Q0 Z 1þ2 H 1þ2 F 1þ2 b 2 : ð45þ Now consider D n2 : Assumption 6(b),(c) imply tat ED n2 ¼ kq0 Z 1þ2 H 1þ2 ðn 1 þ n 2 Þ 1=2 Hn1þ2 0 Wu n1þ2 ði n kw n Þ 1 n3 ½X nb 1 þ W n J n b 2 Š ð46þ! kq 0 Z 1þ2 H 1þ2 ½L 1þ2 b 1 þ S 1þ2 b 2 Š wic is a finite (s? r) 9 1 vector. Te variance of D n2 was computed in te proof of Teorem 2. Te expression for te variance of D n2 is given in (42) and was sown to converge to zero since! 0. Hence by Cebycev s inequality, n 3 ðn 1 þn 2 Þ D n2!p kq 0 Z 1þ2 H 1þ2 ½L 1þ2 b 1 þ S 1þ2 b 2 Š Pulling results togeter, it now follows from (37), (45), and (47) tat ^Z 1þ2 ^Z 1þ2 D n! P ^Z 1þ2 ^Z 1þ2 Q 0 Z 1þ2 H 1þ2 F 1þ2 b 2 þkq 0 Z 1þ2 H 1þ2 ðl 1þ2 b 1 þ S 1þ2 b 2 Þ i ð47þ ð48þ Teorem 3 follows from (30) (32), (34), (37), (45), (47), and (48).

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