Estimation of spatial autoregressive panel data models with xed e ects

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1 Estimation of satial autoregressive anel data models with xed e ects Lung-fei Lee Deartment of Economics Ohio State University l eeecon.ohio-state.edu Jihai Yu Deartment of Economics University of Kentucky jihai.yuuky.edu March 4, 8 Abstract This aer establishes asymtotic roerties of quasi-maximum likelihood estimators for xed e ects SAR anel data models with SAR disturbances where the time eriods T and/or the number of satial units n can be nite or large in all combinations excet that both T and n are nite. A direct aroach is to estimate all the arameters including xed e ects. We roose alternative estimation methods based on transformation. For the model with only individual e ects, the transformation aroach yields consistent estimators for all the arameters when either n or T are large, while the direct aroach does not yield a consistent estimator of the variance of disturbances unless T is large, although the estimators for other arameters are the same as those of the transformation aroach. For the model with both individual and time e ects, the transformation aroach yields consistent estimators of all the arameters when either n or T are large. When we estimate both individual and time e ects directly, consistency of the variance arameter requires both n and T to be large and consistency of other arameters requires n to be large. JEL classi cation: 3; 3; R5 Keywords: Satial autoregression, anel data, Fixed e ects, Time e ects, Quasi-maximum likelihood estimation, onditional likelihood Lee acknowledges nancial suort for his research from NSF under Grant No. SES-594

2 Introduction Satial econometrics deals with the satial interactions of economic units in cross-section and/or anel data. To cature correlation among cross-sectional units, the satial autoregressive (SAR) model by li and Ord (973) has received the most attention in economics. It extends autocorrelation in times series to satial dimensions and catures interactions or cometition among satial units. Early develoment in estimation and testing is summarized in Anselin (988), ressie (993), Kelejian and Robinson (993), and Anselin and era (998), among others. The satial correlation can be extended to anel data models (Anselin, 988). altagi et al. (3) consider the seci cation test of the satial correlation in a anel regression with error comonent and SAR disturbances. Kaoor et al. (7) rovide a rigorous theoretical analysis of a anel model with SAR disturbances which incororate error comonents. altagi et al. (7) generalize altagi et al. (3) by allowing for satial correlations in both individual and error comonents such that they might have di erent satial autoregressive arameters, which encomasses the satial correlation seci cations in altagi et al. (3) and Kaoor et al. (7). Instead of random e ect error comonents, an alternative seci cation for anel data model assumes xed e ects. The xed e ects seci cation has the advantage of robustness in that the xed e ects are allowed to correlate with included regressors in the model (Hausman, 978). Yu et al. (6, 7) and Yu and Lee (7) consider the satial correlation in a dynamic anel data setting, where the data generating rocesses (DGs) are seci ed to be, resectively, stationary, artially nonstationary and nonstationary. For anel data models with xed individual e ects, when the time dimension T is xed, we are likely to encounter the incidental arameters roblem discussed in Neyman and Scott (948). This is because the introduction of xed e ects increases the number of arameters to be estimated. In a linear anel regression model or a logit anel regression model with xed individual e ects, the xed e ects can be eliminated by the method of conditional likelihood when e ective su cient statistics can be found for each of the xed e ects. For those anel models, the time average of the deendent variables rovides the su cient statistic (see Hsiao, 986). For the linear anel regression model with xed e ects, the direct maximum likelihood (ML) aroach will estimate jointly the common arameters and xed e ects. The corresonding ML estimates (MLEs) of the regression coe cients are known as the within estimates, which haen to be the conditional likelihood estimates conditional on the time means. For the SAR anel data models with individual e ects, similar ndings of the direct ML aroach will be shown in this aer. This direct estimation aroach will yield consistent estimates for the satial and regression coe cients excet for the variance of the disturbances However, e ective su cient statistics might not be available for many other models. The well-known examle is the robit anel regression model, where the time average of the deendent variables does not rovide the su cient statistic even though robit and logit models are close substitutes (see hamberlain, 98).

3 when T is small (but n is large). However, for SAR anel models with time e ects, the direct estimation aroach will be shown to be inconsistent for all arameters when n is small (but T is large). The inconsistent estimates are consequences of the incidental arameters (Neyman and Scott, 948). In this aer, in order to avoid the incidental arameters roblem, we suggest alternative estimation methods. y using the data transformation (I T T l T lt ) to eliminate the individual e ects, the transformed disturbances are uncorrelated although not i:i:d: in general. The transformed equation can be estimated by the quasi-maximum likelihood (QML) aroach. For the more general model with both individual and time xed e ects, one may combine the transformation (I n n l nl n) with the transformation (I T T l T l T ) to eliminate both the individual and time xed e ects. y exloring the generalized inverse of the transformed equation, one may end u with a QML aroach for the transformed model 3. anel regression models with SAR disturbances have been recently considered in the literature. model considered in altagi et al. (3) is Y nt = X nt + c n + U nt ; U nt = W n U nt + V nt ; t = ; ; :::; T, where elements of V nt are i:i:d: (; ), c n is an n vector of individual error comonents and the satial correlation is in U nt. The A di erent seci cation has been considered in Kaoor et al. (7) with Y nt = X nt + U + nt and U + nt = W n U + nt + d n + V nt ; t = ; ; :::; T, where d n is the vector of individual error comonents. Kaoor et al. (7) roose method of moment (MOM) rocedure for the estimation of and the variance arameters of d n and V nt. The two anel models are di erent in terms of the variance matrices of the overall disturbances. The variance matrix in altagi et al. (3) is more comlicated and its inverse is comutationally demanding; the variance matrix in Kaoor et al. (7) has a secial attern and its inverse can be easier to comute. altagi et al. (7) allow for satial correlations in both individual and error comonents where they might have di erent satial autoregressive arameters. oth altagi et al. (3) and altagi et al. (7) have emhasized on the test of satial correlation in their models. With the xed e ects seci cation, these anel models can have the same reresentation. y the transformation (I n W n ), the DG of Kaoor et al. (7) becomes Y nt = X nt + c n + U nt where c n = (I n W n ) d n and U nt = U + nt (I n W n ) d n. The U nt = W n U nt + V nt forms a SAR rocess. y regarding (I n W n ) d n as a vector of unknown xed e ect arameters, these two equations are identical to a linear anel regression with xed e ects and SAR disturbances. Hence, to generalize altagi et al. (3), altagi et al. (7) and Kaoor et al. (7), where the satial e ects are in the disturbances, and to generalize the SAR anel model where the satial e ects are in the regression equation, we are going to consider the estimation of the SAR anel model with both satial lag and satial disturbances. We allow that the time eriods T and/or the number of satial units n can be nite or large in all combinations excet When a dynamic e ect is considered into the SAR anel data, we will have an initial condition roblem which will cause the inconsistency of the direct likelihood estimates for all the arameters unless T is large ( see Yu et al, 6, 7 and Yu and Lee (7)). 3 The use of (I T T l T lt ) to eliminate time xed e ects has been considered in Lee and Yu (7a) for a satial dynamic anel model with large T. In a grou setting with grou xed e ects, a similar transformation can eliminate the grou e ects (Lee et al., 8).

4 that both T and n are nite. In this aer, we ay secial attention to the model with individual e ects when n is large but T is small. On the other hand, for the model with time e ects, the secial interest is on the model with large T but small n. This aer is organized as follows. In Section, the model with individual xed e ects is introduced and the data transformation rocedure is roosed. We then establish the consistency and asymtotic distribution of the QML estimator of the transformation aroach. The direct ML aroach is discussed in Section 3 where the individual e ects are estimated directly. Section 4 generalizes the model to include both individual and time e ects. After the individual e ects are eliminated, we can further eliminate the time e ects and the asymtotics are derived. Alternatively, we can estimate the transformed time e ects directly, or estimate both e ects directly, both of which are discussed in Section 5. Simulation results are reorted in Section 6 to comare di erent aroaches. Section 7 concludes the aer. roofs are collected in the Aendix. Transformation Aroach The SAR anel model with SAR disturbances where we have individual e ects is Y nt = W n Y nt + X nt + c n + U nt ; U nt = M n U nt + V nt ; t = ; ; :::; T, (.) where Y nt = (y t ; y t ; :::; y nt ) and V nt = (v t ; v t ; :::; v nt ) are n column vectors and v it is i:i:d: across i and t with zero mean and variance, W n is an n n satial weights matrix, which is redetermined and generates the satial deendence among cross sectional units y it, X nt is an n k X matrix of nonstochastic regressors, and c n is an n column vector of xed e ects. In anel data models, when T is nite, we need to take care of the incidental arameters roblem. In dynamic anel data, the rst di erence or Helmert transformation can be made to eliminate the individual e ects (see Anderson and Hsiao (98) and Arellano and over (995) among others). In this aer, we use an orthogonal transformation which includes the Helmert transformation as a secial case. Our asymtotic results are obtained where T and/or n can be nite or large in all combinations excet that both T and n are nite 4. De ne S n () = I n W n and R n () = I n M n for any and. At the true arameter, S n = S n ( ) and R n = R n ( ). Then, resuming S n and R n are invertible, (.) can be rewritten as Y nt = S n X nt + S n c n + S n R n V nt. (.) For our analysis of the asymtotic roerties of estimators, we make the following assumtions: Assumtion. W n and M n are nonstochastic satial weights matrices and their diagonal elements satisfy w n;ii = and m n;ii = for i = ; ; ; n. 4 We do not have an exact nite small samle theory for the estimators with both n and T being nite. 3

5 Assumtion. The disturbances fv it g, i = ; ; :::; n and t = ; ; :::; T; are i:i:d across i and t with zero mean, variance and E jv it j 4+ < for some >. Assumtion 3. S n () and R n () are invertible for all and. Furthermore, and are comact, is in the interior of and is in the interior of. Assumtion 4. The elements of X nt are nonstochastic and bounded 5, uniformly in n and t. Also, under T the setting in Assumtion 6, the limit of ~ nt t= Xnt X ~ nt exists and is nonsingular. 6 Assumtion 5. W n and M n are uniformly bounded in row and column sums in absolute value (for short, U) 7. Also S n () and R n () are U 8, uniformly in and. Assumtion 6. () n is large, where T can be nite or large; or, () T is large, where n can be nite or large. Assumtion is a standard normalization assumtion in satial econometrics. This assumtion hels the interretation of the satial e ect as self-in uence shall be excluded in ractice. Assumtion rovides regularity assumtions for v it and our analysis is based on i.i.d. disturbances. If there are unknown heteroskedasticity, the MLE (QMLE) would not be consistent. onsistent methods such as the GMM in Lin and Lee (5) and that in Kelejian and rucha (7) may be designed for the model. Invertibility of S n () and R n () in Assumtion 3 guarantees that (.) is valid. Also, comactness is a condition for theoretical analysis. In many emirical alications, each of the rows of W n and M n sums to, which ensures that all the weights are between and. When W n and M n are row normalized, it is often to take a comact subset of (-,) as the arameter sace. When exogenous variables X nt are included in the model, it is convenient to assume that the exogenous regressors are uniformly bounded as in Assumtion 4. Assumtion 5 is originated by Kelejian and rucha (998, ) and also used in Lee (4, 7). That W n, M n, Sn () and R n () are U is a condition that limits the satial correlation to a manageable degree. Assumtion 6 allows three cases: (i) both n and T are large; (ii) T is xed and n is large; (iii) n is xed and T is large. For (ii), we are interested in the short anel data case in contrast to the case where T needs to be large in other studies, e.g., Hahn and Kuersteiner () and Yu et al. (6). When n is large and T is nite, the incidental arameter roblem may aear so that careful estimation methods need to be designed. However, our suggested transformation aroach for the estimation of (.) is general and it may also aly to the cases (i) and (iii) where T can be large. 5 If X nt is allowed to be stochastic and unbounded, aroriate moment conditions can be imosed instead. 6 For notational uroses, we de ne Y ~ nt = Y nt YnT and Y ~ n;t = Y n;t YnT; for t = ; ; ; T where Y nt = T T t= Ynt and Y nt; = T T t= Y n;t. Similarly, we de ne X ~ nt = X nt XnT and V ~ nt = V nt VnT. 7 We say a (sequence of n n) matrix n is uniformly bounded in row and column sums if su n k nk < and su n k nk <, where k nk = su n in j= j ij;nj is the row sum norm and k nk = su n jn i= j ij;n j is the column sum norm. 8 This assumtion has e ectively ruled out some cases, and, hence, imosed limited deendence across satial units. For examle, if n = =n under n!, it is a near unit root case for a cross sectional satial autoregressive model and Sn will not be U (see Lee and Yu (7b)). 4

6 . Data Transformation and onditional Likelihood Let [F T;T ; T l T ] be the orthonormal matrix of the eigenvectors of J T = (I T T l T l T ), where F T;T the T (T ) eigenvector matrix 9 corresonding to the eigenvalues of one and T l T is the T -dimensional column vector of ones. For any n T matrix [Z n ; ; Z nt ] where each Z nt, t = ; ; T, is a T - dimensional column vector, we de ne the corresonding transformed n (T ) matrix [Z n; ; Z n;t ] = [Z n ; ; Z nt ]F T;T. Denote X nt = [X nt;; X nt;; ; X nt;k X ]. Then, (.) imlies ecause E. V n V n. V n;t V n;t Ynt = W n Ynt + Xnt + Unt; Unt = M n Unt + Vnt, t = ; ; T. (.3) A = (FT;T I n) V n. V nt A and v it is i:i:d:, we have A (V n; ; V n;t ) = (F T;T I n)(f T;T I n ) = (F T;T F T;T I n ) = I n(t ). Hence, vit s are uncorrelated for all i and t (and indeendent under normality) where v it is the ith element of V nt. Denote = ( ; ; ; ) and = ( ; ; ). At the true value, = ( ; ; ; ) and = ( ; ; ). The likelihood function of (.3) as if the disturbances were normally distributed, is ln L n;t () = n(t ) ln where V nt() = R n ()[S n ()Y nt n(t ) ln + (T )[ln js n ()j + ln jr n ()j] is X T t= V nt()v nt(), (.4) X nt]. Thus, V nt = V nt( ). The QMLE ^ nt is the extremum estimator derived from the maximization of (.4). For any n-dimensional column vectors nt and q nt, as T t= ntq nt = ( n; ; nt )(F T;T I n )(F T;T I n )(q n; ; q nt ) = ( n; ; nt )(J T I n )(q n; ; q nt ) = T t= ~ nt~q nt by using (~ n ; ; ~ nt ) = ( n ; ; nt )J T, (.4) can be rewritten as ln L n;t () = n(t ) ln where ~ V nt () = R n ()[S n () ~ Y nt n(t ) ln + (T )[ln js n ()j + ln jr n ()j] X T ~V nt() V ~ nt (), t= (.5) ~ Xnt ]. From (.5), the rst and second order derivatives of the likelihood function are (A.) and (A.) in Aendix A.. At true, they are (A.3) and (A.4). We note that the likelihood function in (.5) has a conditional likelihood interretation. It is the conditional likelihood conditional on Y nt, which is a su cient statistic for c n under normality. This is so as follows. (.) imlies that 9 A secial selection of F T;T gives rise to the Helmert transformation where V nt is transformed to ( T t T t+ )= [V nt T t (V n;t+ + + V nt )], which is of articular interest for dynamic anel data models. 5

7 Y nt = W n YnT + X nt + c n + U nt with U nt = M n UnT + V nt and ~ Y nt = W n ~ Ynt + ~ X nt + ~ U nt with ~ U nt = M n ~ Unt + ~ V nt. As ~ V nt, t = ; ; T, are indeendent of V nt under normality, the likelihood in (.5) corresonds to the density function of ~ Y nt, t = ; ; T.. Asymtotic roerties For the likelihood function (.5) divided by the e ective samle size n(t value function is Q n;t () = E max cn n(t Q n;t () = = ) ln L n;t (; c n ), which is ), the corresonding exected n(t ) E ln L n;t () (.6) ln ln + n [ln js n()j + ln jr n ()j] n(t ) E X T ~V nt() V ~ nt (). t= To show the consistency of ^ nt, we need the following uniform convergence result. laim Let be any comact arameter sace of. Under Assumtions -6, n(t ) ln L n;t () Q n;t ()! uniformly in and Q n;t () is uniformly equicontinuous for. roof. See Aendix A.. For local identi cation, a su cient condition (but not necessary) is that the information matrix ;nt, where ;nt = E ln L n;t ( ) n(t ), is nonsingular and E ln L n;t () n(t ) has full rank for any in some neighborhood N( ) of (see Rothenberg (97)). The ;nt is derived in (A.4) of Aendix A. and its nonsingularity is analyzed in Aendix A.3. While the conditions for the nonsingularity of the information matrix rovide local identi cation, the conditions in the following assumtion are global ones. Denote H nt () = X T n(t ) ( X ~ nt ; G n Xnt ~ ) Rn()R n ()( X ~ nt ; G n Xnt ~ ), t= n() = n tr[(r n()r n ) (R n ()R n )], n(; ) = n tr[(r n()s n ()S n R n ) (R n ()S n ()S n R n )]. Assumtion 7. Either (a) the limit of H nt () is nonsingular for each ossible in and the limit of n ln Rn Rn n ln n ()Rn () Rn () is not zero for 6= ; or (b) the limit of n ln R n S n S is not zero for (; ) 6= ( ; ). n Rn n ln n (; )R n () S n () S n ()Rn () When n is nite and T is large, this inequality becomes n ln j R n Rn j n ln j n ()R n () Rn ()j 6=. The inequality will be n ln j R n Sn Sn Rn j n ln j n(; )Rn () Sn () Sn ()Rn ()j 6= when n is nite and T is large. When M n = W n and 6=, this condition would not be satis ed as ( ; ) and ( ; ) could not be distinguished from each other. Identi cation will rely on either Assumtion 7 (a) or extra information on the order of magnitudes of and. 6

8 This assumtion states the identi cation conditions of the model which generalize those for a cross section SAR model in Lee and Liu (6) to the anel case. The art (a) of Assumtion 7 reresents the ossible identi cation of and through the deterministic art of the reduced form equation of (.3) and the identi cation of and from the SAR rocess of Unt in (:3). The art (b) of Assumtion 7 rovides identi cation through the SAR rocess of the reduced form of disturbances of Ynt. The global identi cation and consistency are shown in the following theorem. Theorem Under Assumtions -7, is globally identi ed and, for the extremum estimator ^ nt from (.5), ^ nt!. roof. See Aendix A.4. derived The asymtotic distribution of the QMLE ^ nt can be derived from the Taylor exansion of ln L n;t (^ nt ) around. At, the rst order derivative of the likelihood function involves both linear and quadratic functions of V ~ ln L nt and is derived in (A.3). The variance matrix of n;t ( ) n(t ) is equal to ln L n;t ( ) E ln L n;t ( ) n(t ) = ;nt + ;n, kx k X and ;n = n 4 34 kx n i= G n;ii 4 n n kx n i= G n;ii H n;ii n i= H n;ii kx ntr G A is a symmetric matrix with 4 n ntrh n 4 4 being the fourth moment of v it, where G n;ii is the (i; i) entry of G n, H n;ii is the (i; i) entry of H n, Gn is a matrix transformed from G n as de ned in Aendix A. after (A.4). When V nt are normally distributed, ;n = because = for a normal distribution. Denote as the limit of ;nt and as ln L the limit of ;n, then, the limiting variance matrix of n;t ( ) n(t ) is equal to +. The ln L asymtotic distribution of n;t ( ) n(t ) can be derived from the central limit theorem for martingale di erence arrays. Denote n = G n trg n n I n and D n = H n trh n n I n. Assumtion 8. The limit of n tr( s n s n)tr(d s nd s n) tr ( s nd s n) is strictly ositive. 3 Assumtion 8 is a condition for the nonsingularity of the limiting information matrix (see Aendix A.3). When the limit of H nt is singular, as long as the limit of n tr( s n s n)tr(d s nd s n) strictly ositive, the limiting information matrix a small neighborhood of. 4. tr ( s nd s n) is is still nonsingular. Also, its rank does not change in ln L laim Under Assumtions -6 and 7(a); or -6, 7(b) and 8, n;t ( ) n(t ) ln L When fv it g, i = ; ; :::; n and t = ; ; :::; T; are normal, n;t ( ) d n(t )! N(; ). d! N(; + ). When T is nite, we can use the central limit theorem in Kelejian and rucha (). When T is large, we can use the central limit theorem in Yu et al. (6). 3 When n is nite and T is large, Assumtion 8 is n tr( s n s n)tr(d s nd s n) tr ( s nd s n) >. 4 See (.) in Yu et al. (6) for the case T is large. When T is nite, it still holds according to Lee (4). 7

9 roof. See Aendix A.5. Also, under Assumtions -7, we have ln L n;t () ln L n;t ( ) n(t ) n(t ) = k k O () and = O. 5 ombined with laim, we have the following theorem n(t ) ln L n;t ( ) Q n;t ( ) n(t ) for the distribution of ^ nt. Theorem Under Assumtions -6 and 7(a); or -6, 7(b) and 8, for the extremum estimator ^ nt derived from (.5), n(t )(^nt ) d! N(; ( + ) ), (.7) Additionally, if fv it g, i = ; ; :::; n and t = ; ; :::; T; are normal, (.7) becomes n(t )(^ nt ) d! N(; ). roof. See Aendix A.6. Hence, after the data transformation to eliminate the individual e ects, the QMLE is consistent and asymtotically normal when either n or T are large. 3 The Direct Aroach For the estimation of the linear anel regression model with xed individual e ects, the ML aroach which estimates the xed e ects directly rovides consistent estimates of the regression coe cients, which are known as the within estimates. For the satial anel model with xed individual e ects, one may wonder whether or not the ML aroach will yield consistent estimates when T is small. As we will see below, this direct aroach will yield the same consistent estimator of the transformation aroach for = ( ; ; ) ; however, the estimator of is inconsistent unless T is large. 3. The Likelihood Function The likelihood function for the model before transformation (.) is ln L d n;t (; c n ) = nt ln nt ln + T [ln js n ()j + ln jr n ()j] X T t= V nt()v nt (), (3.) where V nt () = R n ()[S n ()Y nt X nt c n ]. We can estimate c n directly and have the asymtotic analysis on the estimator of via the concentrated likelihood function. Using the rst order condition that ln Ld n;t (;cn) c n = R n() T t= V nt(), we have ^c nt () = T T t= (S n()y nt X nt ) and the concentrated likelihood is ln L d n;t () = nt ln nt ln + T [ln js n ()j + ln jr n ()j] X T ~V nt() V ~ nt (), (3.) t= 5 See (.7) and (.8) in Yu et al. (6) for the case T is large. When T is nite, it still holds according to Lee (4). 8

10 with ~ V nt () being the same one in (.5). One may comare the concentrated likelihood function in (3.) with the likelihood function from the transformation aroach in (.5). We see that the di erence is on the use of T in (3.) but (T ) in (.5). For large T, the two functions can be very close to each other. Therefore, we may exect that the estimates of from these two aroaches could be asymtotically equivalent when T is large. The interesting comarison is for the case where T is nite. 3. Asymtotic roerties For (3.), we can further concentrate out and and focus on (; ). Denote ^ d h T nt (; ) = ~ t= XntR n()r n () X ~ i h T ~ nt t= XntR n()r n ()S n () Y ~ i nt, hs n () Y ~ nt Xnt^d ~ nt (; )i R n ()R n () ^ d nt (; ) = nt T t= The concentrated log likelihood function of (; ) is ln L d n;t (; ) = nt (ln() + ) nt hs n () ~ Y nt ~ Xnt^d nt (; ) i. ln ^d nt (; ) + T [ln js n ()j + ln jr n ()j]. (3.3) We can comare it with the concentrated likelihood function from (.5) where the corresonding estimates are ^ nt (; ) = ^ nt (; ) = h T ~ t= XntR n()r n () X ~ i h T ~ nt t= XntR n()r n ()S n () Y ~ i nt, h S n () Y ~ nt Xnt^nT ~ (; )i R n ()R n () n(t ) T t= h S n () ~ Y nt ~ Xnt^nT (; ) and the ^ nt (; ) for the transformed aroach is consistent even when T is small ( n goes to in nity). The concentrated log likelihood function of (; ) from (.5) is ln L n;t (; ) = n(t ) (ln() + ) n(t ) i, ln ^ nt (; ) + (T )[ln js n ()j + ln jr n ()j]. (3.4) Note that ^ nt (; ) is the same as ^ d nt (; ), and ^ d nt (; ) = T ^ nt (; ). Equation (3.3) can be rewritten as ln L d n;t (; ) = nt T (ln() + ln T T + ) nt ln ^ nt (; ) + T [ln js n ()j + ln jr n ()j]. y comaring (3.3) and (3.4), we can see that they will yield the same maximizer (^ nt ; ^ nt ). As ^ d nt (; ) has the same exression as ^ nt (; ), we can conclude that the QMLE of = ( ; ; ) from this direct aroach will yield the same consistent estimate as the transformation aroach. However, the estimation of from the direct aroach will not be consistent unless T is large, which can be seen from ^ d nt (; ) and ^ nt (; ). 6 Hence, the ML estimation of the satial anel model with xed individual e ects shares some common features on their estimates with those of the ML estimation of the linear anel regression model with xed e ects. 7 6 Note that, for the linear anel regression model with xed e ects, while the within estimates of the regression coe cients are consistent, the corresonding MLE of is not, which is the consequence of the incidential arameters roblem (Neyman and Scott 948). 7 As the bias of the direct estimate of is due to the degree of freedom (T ) instead T, one may easily correct the biased estimate to a bias corrected estimate. The bias corrected estimator will become the conditional likelihood estimator in this model. 9

11 4 A General Model With Time E ects: Transformation Aroach oth altagi et al. (3) and Kaoor et al. (7) focus on models with only individual e ects. While in the anel data literature, there are also two way error comonent regression models where we have not only unobservable individual e ects but also unobservable time e ects (See Wallace and Hussain (969), Amemiya (97), Nerlove (97) and Hahn and Moon (6), etc). Hence, it is natural to generalize the model to include both individual e ects and time e ects. This would be useful for emirical alications where the time dummy e ects might be imortant and should be taken into account, for examle, in growth theory and regional economics (see Ertur and Koch (7) and Foote (7) for recent emirical alications of anel data models with both time dummy e ects and satial e ects). Hence, we generalize (.) to Y nt = W n Y nt + X nt + c n + t l n + U nt ; U nt = M n U nt + V nt, t = ; ; :::; T, (4.) where t is the xed time e ects. For (4.), we may rst eliminate the individual e ects by F T;T to (.3), which yields similar Y nt = W n Y nt + X nt + t l n + U nt, U nt = M n U nt + V nt, t = ; ; :::; T, (4.) where [ l n ; l n ; ; T l n] = [ l n ; l n ; ; T l n ]F T;T can be considered as the transformed time e ects. We can make a further transformation to (4.) to eliminate the transformed time e ects. For this further transformation aroach, it is investigated in this section. Alternatively, we can estimate the t directly. Section 5 covers the direct aroach where we will estimate the transformed time e ects directly. Furthermore, we might be interested to investigate the estimators when we estimate both time e ects and individual e ects directly. This is also discussed in Section Data Transformation and the Likelihood Function To eliminate the time dummy e ects, we need W n and M n to be row normalized for analytical urose 8. Also, Assumtion 4 is changed accordingly. Let J n = I n n l nl n be the deviation from the grou mean transformation over satial units. Assumtion. W n and M n are row normalized nonstochastic satial weights matrices. Assumtion 4. The elements of X nt are nonstochastic and bounded, uniformly in n and t. Also, under the setting in Assumtion 6, the limit of nt T t= ~ X ntj n ~ Xnt exists and is nonsingular. Let (F n;n, l n = n) be the orthonormal matrix of eigenvectors of J n where F n;n corresonds to the eigenvalues of ones and l n = n corresonds to the eigenvalue zero. Similar to Lee and Yu (7a), we can transform the n-dimensional vector Y nt to an (n )-dimensional vector Y nt such that Y nt = F n;n Y nt. 8 When W n and M n are not row normalized, we can still eliminate the transformed time e ects; however, we will not have the resentation of (4.3).

12 Hence, (4.) will be transformed into Y nt = (F n;n W n F n;n )Y nt + X nt + U nt ; U nt = (F n;n M n F n;n )U nt + V nt, (4.3) where X nt;k = F n;n X nt;k now (n )(T ). ecause I n ) V n.. V nt and Vnt = Fn;n Vnt. After the transformations, the e ective samle size is V n. Vn;T A = (FT;T F n;n ) V n.. A = (I T Fn;n ). V nt A, we have E. V n V n V n;t V n;t F n;n )(F T;T F n;n ) = (I T I n ) = I (n )(T ). Hence, v it (and indeendent under normality) where v it The likelihood function for (4.3) is is the ith element of V nt. A = (I T F n;n )(F T;T A (V n ; ; V n;t ) = (F T;T s are uncorrelated for all i and t ln L n;t () = (n )(T ) ln (n )(T ) +(T ) ln In F n;n M n F n;n ln + (T ) ln In F n;n W n F n;n TX t= Vnt ()Vnt (), (4.4) where Vnt () = Rn()[(I n Fn;n W n F n;n )Ynt determinant and inverse of (I n F n;n W n F n;n ) are X nt ], R n() = I n F n;n M n F n;n and the In F n;n W n F n;n = ji n W n j, (I n F n;n W n F n;n ) = F n;n (I n W n ) F n;n, and similarly for (I n F n;n M n F n;n ) (see Lee and Yu (7a)). For any n-dimensional column vector nt and q nt, as J n ( n ; ; nt )J T = J n (~ n ; ; ~ nt ), T t= nt qnt = ( n; ; nt )(F T;T F n;n )(FT;T Fn;n )(qn; ; qnt ) = ( n; ; nt )(J T J n )(q n; ; q nt ) = T t= ~ ntj n ~q nt. This imlies that the likelihood function (4.4) is numerically identical to ln L n;t () = (n )(T ) ln (n )(T ) +(T ) ln js n ()j + (T ) ln jr n ()j ln (T ) ln( ) (T ) ln( ) TX ~V nt()j n Vnt ~ (), (4.5) t= where ~ V nt () = R n ()[(I n W n ) ~ Y nt ~ Xnt ]. 9 9 We note that this likelihood function is, in general, not necessarily a conditional likelihood as the samle average over satial units at each t might not be a su cient statistic for the time dummy.

13 4. Asymtotic roerties The rst and second order derivatives of (4.5) are (.) and (.) in Aendix.. From (.) and (.), the score is in (.3) and the information matrix ;nt = E is in (.4). (n )(T ) The following Assumtions rovide conditions for global identi cation. Denote H nt () = ln L n;t ( ) X T (n )(T ) ( X ~ nt ; G n Xnt ~ ) Rn()J n R n ()( X ~ nt ; G n Xnt ~ ), t= n() = n(; ) = n tr[(r n()rn ) J n (R n ()Rn )], n tr[(r n()s n ()Sn Rn ) J n (R n ()S n ()Sn Rn )]. Assumtion 7. Either (a) the limit of H nt () is nonsingular for each ossible in and the limit of n ln Rn J n Rn n ln n ()Rn () J n Rn () is not zero for 6= ; or (b) the limit of n ln Rn Sn J n Sn Rn n ln n (; )Rn () Sn () J n Sn ()Rn () is not zero for (; ) 6= ( ; ). Assumtion 8. The limit of (n ) tr( s n s n)tr(d s nd s n) tr ( s nd s n) is strictly ositive, where n = trj J n G ng n n n I trj n and D n = J n H nh n n n I n. ln L The variance matrix of n;t ( ) (n )(T ) E (n )(T ) ln L n;t ( ) is equal to ln L n;t ( ) = ;nt + ;n, kx k X n where ;n = kx n [(J ngn ) ii ] 4 34 i= 4 n kx n [(J ngn n ) ii (J n H n ) ii ] n [(J n H n ) ii ]. The asymtotics of the transformation aroach with both time and individual e ects eliminated can be i= i= A kx (n )tr(j ng n ) (n )tr(j nh n ) 4 4 obtained similarly as Theorem. Theorem 3 Under Assumtions,,3,4,5,6 and 7 (a); or,,3,4,5,6,7 (b) and 8, for the extremum estimator ^ nt derived from (4.5), (n )(T )(^nt ) d! N(; ( + ) ), (4.6) When n is nite and T is large, this inequality becomes n ln j R n J nrn j n ln j n()rn () J nrn ()j 6=. The inequality will be n ln j R n Sn J nsn Rn j n ln j n (; )Rn () J nsn () Sn ()Rn ()j 6= when n is nite and T is large. When M n = W n and 6=, this condition would not be satis ed as ( ; ) and ( ; ) could not be distinguished from each other. Identi cation will rely on either Assumtion 7 (a) or extra information on the order of magnitudes of and. When n is nite and T is large, Assumtion 8 is (n ) [tr(n s n)tr(d s nd s n) s tr (nd s n)] s >.

14 Additionally, if fv it g, i = ; ; :::; n and t = ; ; :::; T; are normal, (4.6) becomes roof. See Aendix.. (n )(T )(^nt ) d! N(; ): Hence, after the data transformation to eliminate both the individual e ects and time e ects, the QMLE is consistent and asymtotically normal when either n or T are large. 5 A General Model With Time E ects: Direct Aroaches 5. Direct Aroach I: Estimation of Transformed Time E ects Given (4.) where the individual e ects are eliminated and time e ects are still resent, when n! and T might be nite or large, we can estimate the transformed time e ects consistently. Denote T = ( ; ; ; T ), the likelihood function for (4.) is ln L d n;t (; T ) = n(t ) ln n(t ) ln +(T )[ln js n ()j+ln jr n ()j] T X Vnt(; T )Vnt(; T ), where V nt(; T ) = R n()[s n ()Y nt X nt t l n ]. y using the rst order condition, given, the estimate of t is ^ t () = (l nr n()r n ()l n ) l nr n()r n ()(S n ()Y nt X nt). Using R n ()l n = l n, the likelihood function with T concentrated out is t= (5.) ln L d n;t () = n(t ) ln where V nt() = R n ()[S n ()Y nt n(t ) ln + (T )[ln js n ()j + ln jr n ()j] X nt]. For any n-dimensional column vector nt and q nt, as T X Vnt()J n Vnt(); t= (5.) T t= ntj n q nt = ( n; ; nt )(F T;T I n )(I T J n )(F T;T I n )(q n; ; q nt ) = ( n; ; nt )(J T J n )(q n; ; q nt ) = T t= ~ ntj n ~q nt, the likelihood function (5.) is numerically identical to ln L d n;t () = n(t ) ln n(t ) ln + (T )[ln js n ()j + ln jr n ()j] TX ~V nt()j n Vnt ~ (). For the concentrated likelihood function (5.3), the rst and second order derivatives are in (D.) and (D.) in Aendix D.. From Sections and 3, we can see that for the SAR anel data model with only individual e ects, both the transformation aroach and the direct aroach will yield the same consistent estimator of = ( ; ; ). t= (5.3) 3

15 ut the direct aroach will not yield a consistent estimator of as the transformation aroach does, unless T is large. However, for the SAR anel model with both individual and time e ects, this direct aroach will not yield any consistent estimator, unless n is large. For the SAR anel data with both individual and time e ects, one can see the di erence of the two aroaches via their log likelihood functions in (4.5) and (5.3). For the direct aroach, its concentrated likelihood (5.3) does not adjust the degree of freedom in satial units n and also does not adjust the comonents on the determinants of S n () and R n () while the likelihood of the transformed aroach in (4.5) does. These di erences would result in the inconsistent estimates of and in addition to that of. ecause the estimate of will deend on the estimates of and, it would also be inconsistent. To be convincing, the inconsistency of the QMLE with a nite (small) n can be revealed by investigating the robability limit ln L of the normalized gradient vector, d n;t () ln L n(t ) from (D.) and comare it with nt ( ) (n )(T ) from (.) of the transformation aroach. As the one from (.) is zero because the transformation aroach is consistent, the di erences are on the derivatives with, and. For simlicity, let lim and lim denote that at least one of n and T goes to in nity. We have lim L d n;t () n(t ) = lim n ; lim L d n;t () n(t ) = lim L n, and lim d n;t () n(t ) = lim n. These three limits are, in general, not zero unless n is large. When n is nite, does not solve the equation lim ln L d n;t () n(t ) =. The estimator ^ ml which maximizes the concentrated log likelihood ln L d n;t () would solve the normal equa- ln L tion d n;t (^ ml ) n(t ) =. From the asymtotic theory of an extremum (or M) estimation theory, ^ ml would converge in robability to a which solves the limiting equation lim n(t ) ln L d n;t ( ) = (see, e.g., Amemiya (985), h.4). ut 6=, so the estimates from the concentrated likelihood function would not be consistent unless n is large. omared to Section 3, with time e ects included, the direct aroach does not give a consistent estimate of = ( ; ; ) when n is nite (T goes to in nity). 5. Direct Aroach II: Estimation of oth Time and Individual E ects We can also estimate both time e ects and individual e ects directly for (4.). The likelihood function of (4.) is ln L d n;t (; c n ; T ) = nt ln nt ln + T [ln js n ()j + ln jr n ()j] TX Vnt(; c n ; T )V nt (; c n ; T ), where V nt (; c n ; T ) = R n ()[S n ()Y nt X nt c n t l n ]. Using the rst order conditions for t and c n, the likelihood function with both c n and T concentrated out is t= (5.4) ln L d n;t () = nt ln nt ln + T [ln js n ()j + ln jr n ()j] TX ~V nt()j n Vnt ~ (). (5.5) t= For (5.5), the rst and second order derivatives are, resectively, (D.5) and (D.6) in Aendix D.. 4

16 The concentrated likelihood estimates of from (5:5) can be derived from the rst order conditions which set the rst order derivatives in (D.5) to zero. These rst order conditions characterize the concentrated likelihood estimates of the direct aroach. Denote these estimates as ^ nd ; ^ nd ; ^ nd and ^ nd. For the direct estimation of the transformed time e ect in Section 5., their estimates, denoted by ~ nd ; ~ nd ; ~ nd and ~ nd, are characterized by the rst order conditions with (D.). We see that these two sets of rst order conditions are the same excet that the arameter in (D.5) is taken lace by T T in (D.). 3 Thus, it follows that ( ~ nd; ~ nd ; ~ nd ) = (^ nd; ^ nd ; ^ nd ) and ~ nd = T T ^ nd. From Section 5:, the direct estimation of transformed time e ects will yield inconsistent estimators for all the arameters unless n is large. If we are going to estimate both the time e ects and individual e ects directly, the consistency of will require that n is large and the consistency of requires that both n and T are large. 4 6 Monte arlo We conduct a small Monte arlo exeriment to evaluate the erformance of our transformation aroach and the direct ML estimators under di erent settings. We rst check the case that there is individual e ects but no time e ects in the DG (see (.)), where we comare the erformance of the transformation aroach in Section with the direct aroach in Section 3. Then, we check the case that time e ects are also included in the DG (see (4.)), where we comare the transformation aroach in Section 4 with the direct aroaches in Section 5. We rst generate samles from (.): Y nt = W n Y nt + X nt + c n + U nt ; U nt = M n U nt + V nt t = ; ; :::; T, using a = (:; :; :5; ) and b = (; :5; :; ) where = ( ; ; ; ), and X nt ; c n and V nt are generated from indeendent standard normal distributions and both the satial weights matrices W n and M n we use are the same rook matrices 5. We use T = 5, ; 5, and n = 9, 6; 49. For each set of generated samle observations, we calculate the ML estimator ^ nt and evaluate the bias ^ nt. We do this for times to get i= (^ nt ) i. With two di erent values of for each n and T, nite samle roerties of both estimators are summarized in Table. For each case, we reort the bias (ias), emirical standard deviation (E-SD), root mean square error (RMSE) and theoretical standard deviation (T-SD) 6. oth aroaches have the same estimate of = ( ; ; ) while the estimator of by the direct aroach 3 Instead of the rst order conditions, one may also follow the analysis in Section 3 by investigating the two concentrated likelihood functions of (, ) by concentrating out and. 4 For this direct aroach, the asymtotic bias will be of the order O(max(=n; =T )) and we can have bias corrected estimators which have centered normal distributions as long as n=t 3! and T=n 3!. See Aendix D: for more details. 5 We use the rook matrix based on an r board (so that n = r ). The rook matrix reresents a square tessellation with a connectivity of four for the inner elds on the chessboard and two and three for the corner and border elds, resectively. Most emirically observed regional structures in satial econometrics are made u of regions with connectivity close to the range of the rook tessellation. 6 The T-SD is obtained from diagonal elements of the estimated Hessian matrix. 5

17 has a larger bias. The transformation aroach yields a consistent estimator of and the direct aroach does not, which can be seen from the last two columns in Table when T is small. We can see that the iases, E-SDs, RMSEs and T-SDs for estimators of the = ( ; ; ) are small when either n or T are large. Also, T-SDs are similar to E-SDs, which imlies that the Hessian matrix rovides roer estimates for the variances of estimators. Also, when T is larger, the bias of the estimator of by the direct aroach decreases. We then generate samles from (4.): Y nt = W n Y nt + X nt + c n + t l n + U nt ; U nt = M n U nt + V nt, t = ; ; :::; T, using the same n, T, a, b, W n and M n. The X nt ; c n, T = ( ; ; ; T ) and V nt are generated from indeendent standard normal distributions. The nite samle roerties of the estimators are summarized in Table and Table 3, where Table is for the erformance of the estimators using the transformation aroach in Section 4. Table 3 is for the erformance of the estimators using both direct aroaches discussed in Section 5. and 5.. We can see that the bias of the transformation aroach is small. For the aroach that estimates the transformed time e ects directly, the bias is small when n is large, while the bias is large when n is small even though T might be large. For the direct aroach, it has the same estimate of = ( ; ; ) as the aroach that estimates the transformed time e ects directly, while the bias for the estimate of is small only when both n and T are large. This is consistent with the theoretical rediction. Also, when both n and T are large, the biases of all the arameters from three aroaches are small and the RMSEs are reduced. Table -3 here. 7 onclusion In this aer, we consider the estimation of a SAR anel model with xed e ects and SAR disturbances where the time eriods T and/or the number of satial units n can be nite or large in all combinations excet that both T and n are nite. We rst consider the SAR anel model with individual e ects. If T is nite but n is large, we show that a direct ML estimation by estimating jointly all the arameters including the xed e ects will yield consistent estimators excet for the variance of disturbances. These features are similar to the direct ML estimation of the linear anel regression model with xed individual e ects. In this aer, we suggest a transformation aroach, which eliminates the individual xed e ects and can rovide consistent estimates for all the arameters including the variance of disturbances. When the individual e ects are eliminated by taking deviation from time average for each satial unit, the resulted disturbances will be correlated over the time 6

18 dimension and there is linear deendence among the resulted disturbances. The transformation aroach is motivated by a ML aroach which takes into account the generalized inverse of the resulted disturbances. The transformation aroach is shown to be a conditional likelihood aroach if the disturbances were normally distributed. We consider next the SAR model with both individual and time xed e ects. We investigate two ossible direct ML aroaches for the estimation. The rst direct aroach is to transform the data to eliminate the individual e ects and then estimates the remaining arameters including the time e ects by the ML method. The second direct aroach is to estimate both individual and time e ects directly. We show that the rst direct ML aroach will yield inconsistent estimates for all the arameters, unless n is large; and the second direct aroach will yield inconsistent estimates only when both n and T are large. In fact, these two direct ML aroaches rovide identical estimates of the satial e ects and the regression coe cients excet for the estimates of. These results are in contradiction with those of the direct ML estimation of the anel regression models with both individual and time e ects where the regression coe cients can be consistently estimated as long as either n or T is large. onsistent estimation based on transformations is available, where both the individual and time e ects can be eliminated by roer transformations. All the arameter estimates are consistent when either n or T is large. Monte arlo results are rovided to illustrate nite samle roerties of the various estimators with n and/or T being small or moderately large. omared with altagi et al. (3), altagi et al. (7) and Kaoor et al. (7) where random e ects are assumed, the SAR model in this aer considers a xed e ects seci cation. The roosed estimation methods are robust regardless of the di erent seci cations in altagi et al. (3) and Kooor et al. (7), and are comutationally simler than the ML aroach for the estimation of the generalized random e ects model in altagi et al. (7). However, when the individual e ects are random in the true DG, roer methods which take into account the random e ects variance structure can imrove the e ciency of the estimates. Hausman s tye of seci cation test of xed e ects vs random e ects may also be constructed. These may be investigated in the future research. 7

19 Aendices A Transformation Aroach A. The First and Second Order Derivatives For the rst and second order derivatives of (.5), we have T t= (R n() X ~ nt ) Vnt ~ () ln L n;t () = T t= (R n()w n Ynt ~ ) Vnt ~ () (T )trg n () T t= (H n() V ~ nt ()) Vnt ~ () (T )trh n () T 4 t= ( V ~ nt() V ~ nt () n T T ) A, (A.) ln L n;t () = T t= (R n() X ~ nt ) R n () X ~ nt T t= (R n()w n Ynt ~ ) R n () X ~ T nt t= (R n()w n Ynt ~ ) R n ()W n Ynt ~ +(T )tr(g n()) T t= (H n() ~ V nt ()) R n () ~ X nt ) + T ~ t= Vnt()M n Xnt ~ T ~ 4 t= Vnt()R n () X ~ nt + T t= (H n() V ~ nt ()) H n () V ~ nt () +(T )tr(hn()) T 4 t= (H n() V ~ nt ()) Vnt ~ () At true, we have ln L n;t ( ) = n(t ) n(t 4 and the information matrix is equal to ;nt = T ) t= Xnt V ~ nt T n(t ) t= ( G nxnt ) Vnt ~ + T n(t ) t= ( V ~ nth n V ~ T nt T trh n ) T t= (R n()w n Ynt ~ ) nh n () V ~ nt () + T t= (M nw n Ynt ~ ) Vnt ~ () T 4 t= (R n()w n Ynt ~ ) Vnt ~ () n(t ) T t= ( ~ V nt() ~ V nt ()) T n(t ) t= ( V ~ nt V ~ nt n T T ) E n(t ) A. T n(t ) t= ( V ~ nt G ~ T n V nt T trg n ) ln L n;t ( ) = kx k X H nt (kx +) A + kx n tr G s ng n ) kx (kx +) n tr(hs ng n ) n tr(hs nh n ) kx ntr( G n ) ntr(h n) 4 where we denote A s n = A n + A n for any n n matrix A n, G n = W n Sn, Wn = R n W n Rn, Gn = W n (I n Wn ), H n = M n Rn, Xnt = R n Xnt ~ T and H nt = n(t ) t= ( X nt, G nxnt ) ( X nt, G nxnt ). A. roof of laim To rove n(t ) ln L n;t () Q n;t ()! uniformly in in any comact arameter sace : A ; (A.3) (A.4) A (A.), A 8

20 From V ~ nt () = R n ()[S n () Y ~ nt Xnt ~ ], we have ~V nt () Vnt ~ = R n ()[S n () Y ~ nt Xnt ~ ] R n [S n Ynt ~ Xnt ~ ] = R n ()[S n () Y ~ nt Xnt ~ ] R n ()[S n Ynt ~ Xnt ~ ] + R n ()[S n Ynt ~ Xnt ~ ] R n [S n Ynt ~ Xnt ~ ] = R n ()[( )W n Ynt ~ + X ~ nt ( )] + ( )M n [S n Ynt ~ Xnt ~ ] = R n ()[( )W n Ynt ~ + X ~ nt ( )] + ( )H n Vnt ~. Similarly to Lee (4) and Yu et al. (6), we can show that 7 n(t ) TX ~V nt() V ~ nt () t= n(t T ) E X ~V nt() V ~ nt ()! uniformly in. t= Hence, by using the fact that is bounded away from zero in, n(t ) ln L n;t () Q n;t () = uniformly in in. n(t ) TX ~V nt() V ~ nt () t= n(t! T ) E X ~V nt() V ~ nt ()! t= To rove Q n;t () is uniformly equicontinuous in in any comact arameter sace : From (.6), Q nt () = ln ln + n [ln js n()j + ln jr n ()j] n(t ) E X T t= ~V nt() ~ V nt (): The uniform equicontinuity of Q n;t () can be shown similarly to Lee (4) and Yu et al. (6). A.3 Information Matrix We can rove the nonsingularity of the limiting information matrix by using an argument by contradiction (similar to Lee (4)). Denote the limit of ;nt as where ;nt is (A.4), we need to rove that c = imlies c = where c = (c ; c ; c 3 ; c 4 ), c ; c 3 ; c 4 are scalars and c is k X vector. If this is true, then, columns of would be linear indeendent and would be nonsingular. Denote H as the limit T of n(t ) t= Xnt X T nt, H as the limit of n(t ) t= Xnt G nxnt, H = H and H as the limit of T t= ( G n Xnt ~ ) Gn Xnt ~, then 8 n(t ) H H kx kx = H H + lim n! n tr( G s ng n ) lim n! n tr(h s ng n ) lim n! n tr( G n ) kx lim n! n tr(hn s G n ) lim n! n tr(hnh s n ) lim n! n tr(h n) A. kx lim n! n tr( G n ) lim n! n tr(h n) Hence, c = imlies () H c + H c = ; 7 When n is large and T is xed, the derivation is similar to Lee (4) for the cross sectional SAR model. When T is large and n could be nite and large, the derivation is similar to Yu et al. (6). 8 When n is nite and T is large, we do not need the limit before each trace oerator in the entries of. 9

21 () H c + H + lim n! n tr( G s ng n ) c + lim n! n tr(hs ng n ) c 3 + lim n! ntr( G n ) c 4 = ; (3) lim n! n tr(hs ng n ) c + lim n! n tr(hs nh n ) c 3 + lim n! n tr(h n) c 4 =, (4) lim n! n tr( G n ) c + lim n! n tr(h n) c 3 + c 4 =. The rst equation imlies c = (H ) trg H c. Denote n = G n n n I trh n and D n = H n n n so that n tr( G s ng n ) trgn n = n tr(s nn), s n tr(hs nh n ) trhn n = n tr(ds ndn) s and n tr(hs ng n ) trhn n tr G n n = n tr(s nd s n). From the third and fourth equations, we have 4 h n tr(hnh s n )trg n n tr(s ndn)c s + n tr(ds ndn)c s 3 =, tr(hn s G i n )trh n c + n tr(dnd s n)c s 4 =. y eliminating c ; c 3 and c 4, the second equation becomes lim n! n tr(ds ndn) s H H (H ) H + n c = where is nonnegative by the auchy inequality. The H The nonsingularity of follows from Assumtion 7. A.4 roof of Theorem n = 4n tr( s n s n)tr(d s nd s n) tr ( s nd s n) (A.5) H (H ) H is nonnegative by the Schwartz inequality. As E T ~ t= Vnt V ~ nt = n(t ) n(t ) n(t ), at, (.6) imlies E ln L n;t ( ) = ln ln + (T n(t ) )[ln js n j + ln jr n j]. As Y ~ nt = Sn ( X ~ nt + Rn V ~ nt ) and S n ()Sn = I n + ( )G n, we have Denote It follows that where ~V nt () = R n ()[S n ()S n R n ~ V nt + ( )G n ~ Xnt + ~ X nt ( )]: n() = n tr[(r n()r n ) (R n ()R n )], n(; ) = n tr[(r n()s n ()S n R n ) (R n ()S n ()S n R n )]. n(t ) E ln L n;t () n(t ) E ln L n;t ( ) = (ln ln )+ n ln js n()j n ln js nj+ n ln jr n()j n ln jr nj = T ;n (; ; ) T ;n;t (; ; ) T ;n (; ; ) = T ;n;t (; ; ) = (ln ln ) + n ln js n()j X T n(t ) t= n(t ) n ln js nj + n ln jr n()j I n T t= E V ~ nt() V ~ nt () n ln jr nj ( n(; ) ), n ( ~ X nt ( ) + ( )G n ~ Xnt ) R n()r n ()( ~ X nt ( ) + ( )G n ~ Xnt ) o :