QML estimation of spatial dynamic panel data models with time varying spatial weights matrices

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1 QML estimation of satial dynamic anel data models with time varying satial weights matrices Lung-fei Lee Deartment of Economics Ohio State University Jihai Yu Guanghua School of Management eking University November, Abstract his aer investigates the quasi-maximum likelihood estimation of satial dynamic anel data models where satial weights matrices can be time varying. We nd that QML estimate is consistent and asymtotically normal. We investigate marginal imacts of exlanatory variables in this system via sace-time multiliers. Monte Carlo results are reorted to investigate the nite samle roerties of QML estimates and marginal e ects. When satial weights matrices are substantially varying over time, a model misseci cation of a time invariant satial weights matrix may cause substantial bias in estimation. Slowly time varying satial weights matrices would be of less concern. JEL classi cation: C3; C3; R5 Keywords: Satial autoregression, Dynamic anels, ime varying satial weights matrix, Fixed effects, Maximum likelihood Yu ackowledges funding from National Science Foundation of China (Grant No.775) and suort from Center for Statistical Science of eking University.

2 Introduction and Motivation Satial anel data models can deal with cross sectional and dynamic deendences among economic units, allowing for heterogeneity among these unit. hey are generalized from a cross sectional model, where the satial autoregressive (SAR) model by Cli and Ord (973) has received the most attention. Recently, there is much rogress in emirical and theoretical works on satial anel data models. For the static case, satial anel data models can be alied to agricultural economics (Druska and Horrace 4), transortation research (Frazier and Kockelman 5), ublic economics (Egger et al. 5), consumer demand (Baltagi and Li 6), to name a few. For the dynamic case, satial dynamic anel data models can be alied to the growth convergence of countries and regions (Baltagi et al. 7, Ertur and Koch 7), regional markets (Keller and Shiue 7), labor economics (Foote 7), ublic economics (Revelli, ao 5, Franzese 7), and some other elds. For the estimation and statistical inference, theoretical econometrics models are needed. For the random e ects model, Baltagi et al. (3, 7a, 7b), Mutl (6) and Kaoor et al. (7) investigate various seci cations with error comonents. For the xed e ects model, Elhorst (5), Korniotis (), Su and Yang (7), Yu et al. (7, 8) and Lee and Yu (a) study static or dynamic models under various satial structures. Mutl and fa ermayr () and Lee and Yu (b) consider the estimation of satial anel data models with both xed and random e ects seci cations, and roose Hausman-tye seci cation tests. In the current literature of satial anel data models, the satial weights matrix is usually seci ed to be time invariant. Some weights matrices are based on contiguity or distances among regions, but others are constructed with economic/socioeconomic distances or demograhic characteristics. For examle, Case et al. (993) on state sending have weights based on the di erence in the ercentage of the oulation that is black. Brueckner (998) and Brueckner and Saavedrat () in ate weights according to oulation gures in studies of local governments cometitions on economic olicies. Baicker (5) constructs a weights matrix with the degree of oulation mobility between regions. So is Rincke () which exloits information on commuting ows between locations. However, when elements of satial weights matrix are constructed from economic/socioeconomic characteristics of regions (or districts) in a anel or dynamic setting, these characteristics might be changing over time. One may wonder whether models with time varying satial weights can be easily handled and estimated, and whether ignoring time variation in satial weights matrices would have substantial consequences on estimates. hese motivate our investigation on the satial dynamic anel data (SDD) model with time varying satial weights matrices. his aer investigates the maximum likelihood (ML) and quasi-maximum likelihood (QML) estimation of SDD models under the setting of time varying satial weights matrices, which are taken to be exogenous. We also consider the imlication

3 of marginal imacts due to changes in exlanatory variables in the system. he latter can be of interest in emirical alications of such models. his aer is organized as follows. Section introduces the model and resents the likelihood function to be maximized. Section 3 establishes asymtotic roerties of ML and QML estimators. We show that the ML and QML estimates are consistent and asymtotically normal. Comared to the time invariant SDD model (see Lee and Yu (c)), the bias and information matrix of the arameter estimates are aarently di erent. In Section 4, we rovide analysis of marginal imacts of exlanatory variables on outcomes for such a dynamic model. Monte Carlo results for various estimators are rovided in Section 5. We nd that misseci cation with a time invariant weights matrix may lead to inaccuracy in estimation and inference when the satial weights changes are substantial. Slowly changing satial weights matrices are of less concern for estimation and inference. Section 6 concludes the aer. Some lemmas and roofs are collected in the Aendices. he Model and Maximum Likelihood Estimation he model considered is Y nt = W nt Y nt + Y n;t + W n;t Y n;t + X nt + c n + t l n + V nt ; 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 s are i:i:d: across i and t with zero mean and variance. he X nt is an n k x matrix of individually and time varying nonstochastic regressors, c n is an n column vector of individual e ects, and t is the tth element of the xed time e ect vector with l n being n vector of ones. he initial values in Y n are observable. he satial weights matrix W nt is nonstochastic and generates the deendence of y it s across satial units, and it could be time varying. Denote S nt () = I n where A nt = S nt ( I n + W n;t We consider the case that W nt is row-normalized as in common ractice. W nt for an arbitrary at time eriod t. he reduced form of () is Y nt = A nt Y n;t + S nt (X nt + c n + t l n + V nt ); () ). Denote the eigenvalue matrix of W nt by $ nt where $ nt is a diagonal matrix. When we have a time invariant satial weights matrix W n which is diagonalizable with an eigenvalue matrix $ n, the eigenvalue matrix of A n = S n ( I n + W n ) is D n = (I n $ n ) ( I n + $ n ). 3 When satial weights matrices are time variant, the eigenvalues matrix of A nt would not be straightforward because W nt s cannot be diagonalizable with the same eigenvectors, and it is comlicated to see the stability of rocess. If W nt s are all row-normalized, a su cient condition for the eigenvalues of A nt being smaller

4 than is j j < and j j+j j j j <. 4 his is so, because S nt < + j j + j j + = j, and j hence, ka nt k < Snt (j j + j j kw n k ) < j j+j j j j, where kk denotes the row sum matrix norm (see footnote 8). It follows by the sectral radius theorem (see heorem in Horn and Johnson, 99) that all the eigenvalues of A nt are less than one in absolute value. Denote A (h) nt = A nt A n;t A n;t h+ for h = ; ; ::: with A () nt substitution, we have Y nt = X = I n for all t. From (), assuming the existence of in nite sums below, 5 by backward h= A(h) nt S n;t h (c n + X n;t h + t h; l n + V n;t h ) = nt c n + t l n + X nt + U nt, (3) where nt h= A(h) nt S n;t h, tl n nt S n;t h l n = h= t h;( + ) h l n, 6 X nt h= A(h) nt S n;t h X n;t h, and U nt h= A(h) nt S n;t h V n;t h. h= t h;a (h) For the estimation of the model, denote c n and = ( ; :::; ) as arbitrary individual and time e ects vectors and let = ( ; ; ) where = (; ; ). We can write down the likelihood function directly for () in terms of (; c n ; ) and concentrate out c n and. However, this direct aroach has the incidental arameter roblem from both the individual e ects and time e ects, which results in a bias of the order O(max( n ; )) for MLE even for the case with a time invariant W n as in Lee and Yu (c). Although this bias can be eliminated by a bias correction rocedure, it requires the ratio conditions that n 3 and n 3 converge to zero in the limit. Here, the O( ) bias is caused by the individual e ects and the O( n ) bias is caused by the time e ects. he time e ects might be eliminated when W n is row-normalized (see the transformation aroach in Lee and Yu, c) and the resulting equation has a (artial) likelihood function. However, the individual e ects cannot be eliminated in such a fashion due to the dynamic feature. In the current aer, with W nt row-normalized, we will also use the transformation aroach to eliminate the time e ects before estimation. Let (F n;n, l n = n) be the orthonormal matrix of eigenvectors of J n = I n n l nl 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 (c), denoting Y nt = F n;n Y nt and other variables similarly, () can be transformed into Y nt = W nty nt + Y n;t + W n;t Y n;t + X nt + c n + V nt, (4) where Wnt = Fn;n W nt F n;n, Xnt = Fn;n X nt, c n = Fn;n c n, Vnt = Fn;n V nt and Vnt is an (n ) dimensional disturbance vector with zero mean and variance matrix I n. his equation is in the format of a tyical SAR model in anel data, where the time e ects are eliminated and the number of observations is (n ). Denote Z nt = [Y n;t ; W n;t Y n;t ; X nt ] so that S nt Y nt = Z nt + c n + t l n + V nt. he log 3

5 likelihood function for (4) is ln L n; (; c n) = (n ) ln (n ) X ln + ln ji n Wntj t= X t= V nt()v nt(), (5) where V nt() = (I n W nt)y nt Z nt c n and Z nt = F n;n Z nt. As ji n W ntj = ji n W nt j, (I n W nt) = F n;n (I n W nt ) F n;n and V nt()v nt() = V nt()j n V nt () where V nt () = (I n W n )Y nt Z nt c n, the log likelihood function (5) for Y nt can be exressed in terms of Y nt as ln L n; (; c n ) = (n ) ln (n ) X ln ln( )+ ln js nt ()j t= X Vnt()J n V nt (). (6) t= Using rst order conditions, we concentrate out c n (or J n c n ) in (6). Denote Y ~ n;t = Y n;t t= Y n;t, W n;t ^ Y n;t = W n;t Y n;t t= W n;t Y n;t and X ~ nt = X nt t= X nt. hus, Znt ~ = ( Y ~ n;t ; W n;t ^ Y n;t ; Xnt ~ ). he concentrated log likelihood with c n (or J n c n ) concentrated out is ln L n; () = (n ) ln (n ) X ln ln( ) + ln js nt ()j t= X ~V nt()j n Vnt ~ (), (7) t= where V ~ nt () = S ^ nt ()Y nt Znt ~ with Snt ^ ()Y nt = S nt ()Y nt t= S nt()y nt and J n Vnt ~ () = J n [ ~Z nt ~ t l n ] because J n l n =. ^ Snt ()Y nt 3 Asymtotic roerties of the QML Estimate In this section, we consider the roerties of QMLE when both n and are large. 7 analysis of the ML and QML estimators, we assume the following regularity conditions. For our asymtotic Assumtion W nt s are row-normalized nonstochastic satial weights matrices with zero diagonals for all t. Assumtion he disturbances fv it g, i = ; ; :::; n and t = ; ; :::; ; are i:i:d: across i and t with zero mean, variance and E jv it j 4+ < for some >. Assumtion 3 he elements of X nt, c n and are nonstochastic and bounded, uniformly in n and t. Also, lim! t= ~ X ntj n ~ Xnt exists and is nonsingular. Assumtion 4 S nt () is invertible for all t and for all, where the arameter sace is comact and is in the interior of. 4

6 Assumtion 5 W nt s are uniformly bounded in both row and column sums in absolute value (for short, UB), 8 uniformly in t. Also, S nt () s are UB, uniformly in and t. Assumtion 6 su t h= abs(a(h) nt ) is UB, where [abs(a (h) nt )] ij = j(a (h) nt ) ij j. Assumtion 7 n is a strictly increasing function of, and goes to in nity. In Assumtion, we assume that the satial weights matrices are nonstochastic for all t, and self-in uence is excluded. he row-normalization of W nt guarantees that the transformed equation (4) still has an SAR structural form. Assumtion rovides regularity conditions for v it, where the existence of higher than the fourth moment is needed for the central limit theorem related to the quadratic form of V nt s. Assumtion 3 oints out that the regressors of X nt after derivations from both cross section and time averages are asymtotically linear indeendent. his has imlicitly ruled out time or individually invariant regressors. Assumtion 4 guarantees that the model is an equilibrium one for each eriod. he comactness of the arameter sace is convenient for theoretical analysis. Assumtion 5 is originated by Kelejian and rucha (998, ) and also used in Lee (4, 7), which is a condition that limits the satial correlation to a manageable degree. Assumtion 6 is the dynamic stability condition. As W nt is row-normalized for all t, under the assumtion that j j < and j j+j j j j <, we have jja (h) nt jj < ( j j+j j j j ) h for all h = ; ; ; :::. In that case, h= abs(a(h) nt ) is uniformly bounded in row sum norm, uniformly in t. he uniform boundedness in the column sum norm is an additional assumtion. Assumtion 7 seci es that we have a large number of satial units and time eriods. In the following, we investigate asymtotic roerties of MLE and QMLE of model (). We see that the basic analysis in Lee and Yu (c) can be modi ed, where the bias and information matrix formula need to be revised. 3. Consistency For the concentrated likelihood function (7), the rst and second order derivatives are derived in (3) and (3) in Aendix A.. At the true arameter values, denoting G nt = W nt Snt, the score is ln L n; ( ) t= ZntJ n Vnt ~ t= ( G^ nt Z nt + G ~ nt c n ) J n Vnt ~ + t= (( ^G nt V nt ) J n Vnt ~ tr(j n G nt )) C 4 t= ( V ~ A, ntj n Vnt ~ (n ) ) (8) where G^ nt Z nt = G nt Z nt t= G ntz nt and ^G nt V nt = G nt V nt t= G ntv nt. Denoting H c = (n ) t= ( ~ Z nt ; ( ^ G nt Z nt + ~ G nt c n )) J n ( ~ Z nt ; ( ^ G nt Z nt + ~ G nt c n )) with ~ G nt = G nt t= G nt, the 5

7 information matrix ; = E (n ln L n; ( is equal to (see Aendix A.) ; = EH c (kx+3) (kx+)(k x+) B (kx+) (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) (kx+) (n ) t= tr(j ng nt ) 4 C A + O. As is shown in Aendix A.3, the limit of the information matrix is nonsingular if lim! EH c is nonsingular or lim! ( (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) h i tr(j ) ng nt) t= n 6=. he consistency of the QMLE ^ that maximizes (7) can be established similarly to Lee and Yu (c). Denote Assumtion 8 lim! EH c lim ln! (n ) S t= nt() = n tr(s nt S nt()j n S nt ()S nt ). () is nonsingular or nt Snt (n ) t= ln nt()snt () Snt () 6= for 6= : (9) Assumtion 8 is the identi cation condition. If lim! EH c 8 would identify the arameters. is singular, the second art in Assumtion heorem Under Assumtions -8, is globally identi ed and the QMLE ^ ^!. that maximizes (7) has roof. See Aendix B.. 3. Asymtotic Normality he asymtotic distribution of the QMLE ^ can be derived from the aylor exansion ln L n; (^ around, which involves both linear and quadratic functions of V nt. For the asymtotic distribution, we have a decomosition of the score. From (3), we can decomose J n ~ Znt = J n ( ~ Y n;t ; ^ W n;t Y n;t ; ~ X nt ) into J n ~ Znt = J n ~ Z (u) nt (J n U; ; J n W n; U ; ; nkx ), () where Z ~ (u) nt = [(~ n;t c n + X ~ n;t + U n;t ); ( W n;t ^ n;t c n + W n;t ^ X n;t + W n;t U n;t ); Xnt ~ ] with ~ n;t = n;t t= n;t, W n;t ^ n;t = W n;t n;t t= W n;t n;t, Xn;t ~ = X n;t t= X n;t, W n;t ^ X n;t = W n;t X n;t t= W n;t X n;t, U; = t= U n;t and W n; U ; = t= W n;t U n;t. Hence, J n Znt ~ has two comonents: one is J n Z ~ (u) nt, uncorrelated with V nt ; the other is (J nu; ; J n W n; U ; ; nkx ), correlated with V nt when t. herefore, from (8), denoting G nt Z (u) nt = (( G nt ^ ^ n;t c n + G nt ^X n;t + G nt U n;t ); ( G nt W^ n;t n;t c n + 6

8 G nt W n;t ^ X n;t + G nt W n;t U n;t ); G^ nt X nt ), the score can be decomosed into two arts such ln L n; ( ) (n = (n (n and ln L (u) n; ( ) = r n ln L (u) n; () (n 4 (n ) t= where ~ ) t= Z (u) nt J n V ( G ^ nt Z (u) nt + G ~ nt c n ) J n V nt + (n ) t= (V A ntj n G nt V nt tr(j n G nt ), C A (n ) t= (V ntj n V nt (n ) ) n ( U ; ; W n; U ; ; nkx ) J n V n (G U n; ; G W n; U ;, nkx ) ) J nv + n V J nv As is shown in Aendix A.4, the variance matrix of (n E ln L (u) n; () is equal ln L (u) n; ( ln L (u) n; (! ) (n = ; + ;n + O n G V JnV, where ; is in (9) and ;n is in (33). When V nt are normally distributed, ;n = because 3 = () C A. (3) and = where 3 and 4 are the third and fourth moments of v it. Denote = lim! ln L and = lim! ;n, then, lim! E (u) n; ln L (n (u) n; () (n = +. he asymtotic distribution of ln L (u) n; () can be derived from the central limit theorem for martingale q n a ;n + O( n ) + di erence arrays (see Lemma 5). For the term, from Lemma, we have = O ( ) where a ;n = O() and 9 a ;n = 4 (n ) tr J n (n ) tr J n t= (n ) tr J n t= (n ) tr t t= J n t= h= G n;t+h()w n;t+h A t ) h= A(h n;t+h t (h ) n;t+h kx t (h ) h= W n;t+h A () h= G n;t+h()a n;t+h (h ) n;t+h () () Snt () Snt () () Snt () Snt () + + (n ) tr(j n t= G nt()) (4) 3 5. C A Assumtion 9 If lim! EH c lim! is singular, X tr(g (n ) nt J n G nt ) + tr((j n G nt ) ) X t= t=! tr(j n G nt ) 6= : n 7

9 As is derived in Aendix A.3, the information matrix would be nonsingular if lim! EH c is nonsingular or Assumtion 9 holds. heorem Under Assumtions -9, the QMLE ^ that maximizes (7) has r r r!! n n (n ) (^ ) + b ; + O max 3 ; d! N(; ( + ) ), (5) where b ; = ; a ;n is O(). roof. See Aendix B.. From heorem, we have three cases for the QMLE ^. When n!, we have a centered distribution where (n ) (^ ) d! N(; ( + ) ). When n! k <, the QML estimator ^ has the bias b ; and the con dence interval is not roerly centered; we see that (n ) (^ ) + d kb;! N(; ( + ) ). When n!, (^ ) + b ;!. In all these cases, we see that the bias term with b ; is of order O( ), which will vanish as tends to in nity. he estimator ^ is still consistent as in heorem even with the resence of such a bias. his bias occurs for the anel dynamic model due to the initial value and the incidental arameter roblems, and it does not aear in a cross sectional SAR model as these two roblems are not resent. For a static satial anel model, the initial value roblem is not resent, and the incidental arameter roblem is of less concern (see Lee and Yu, d). An analytical bias reduction rocedure can correct the bias of the estimate ^. De ne the bias corrected estimator as ^ = ^ where ^ is the QMLE that maximizes (7) and ^B h = from (4). Assumtion t h= ) A(h nt () and t h= uniformly in a neighborhood of and t. (h nt ^B ; a ;n i =^ (6) with ; from (9) and a ;n are uniformly bounded in either row or column sums, Assumtion assures is uniformly bounded in a neighborhood of. heorem 3 Under Assumtions -, if n 3!, (^ ) d! N(; + ). roof. See Aendix B.. o estimate individual e ects and time e ects, we imose t= t = to avoid the un-identi cation of c i; and t because c i; + t = (c i; + ) + ( t ) for an arbitrary. As both n and are large, the 8

10 individual e ects and time e ects can be estimated consistently. Denoting ^r nt = S n (^ )Y nt the estimate for r nt = c n + t l n + V nt (= S nt Y nt ^c n = t= ^r nt and that for t is ^ t; = n l n(^r nt ^c n) = n l n(^r nt the individual e ects and time e ects are Z nt^ Z nt ). With t= t =, the estimate for for c n is as t= ^r nt). hus, the estimates for ^c n = X t= (S nt (^ )Y nt Z nt^ ); (7) and ^ t; = ^ n l n( S nt (^ )Y nt ~ Znt^ ); for t = ; :; ; ; :. (8) As is shown in Aendix B., for each element of the estimate ^c n, ^c i;, we have ^c i; c i; d! N(; ) and they are asymtotically indeendent with each other. hus, the estimators of xed e ects are consistent and asymtotically centered normal. Similarly, for estimate ^ t; of each time e ect, we have n ^ d! t; t N(; ) and they are asymtotically indeendent with each other. 4 Sace-ime Multiliers and Marginal Imact he (3) exresses the model in terms of a sace-time multilier (Anselin et al. 8), which seci es how the joint determination of the deendent variables is a function of both satial and time lags of exlanatory variables and disturbances at all locations of satial units. his equation is useful for calculating marginal e ects of changes of exogenous variables on outcomes over time and across satial units. We are interested in the imact analysis on the future exected outcomes. For analytical urose, we assume that the time e ects have zero means so that E( t ) =. he regressors X nt and weights matrices W nt s are assumed to be given. From (3), we have the following equation in terms of exectation of Y nt given the ast u to (the last samling eriod), E(Y nt ) = A (t ) nt Y + X t h= A (h) nt S n;t h (c n + X n;t h ), where t >. Without loss of generality, consider a single regressor in X nt. As in LeSage and ace (9), we may be interested in an average total (exected) imact resulting from changing the regressor by the same amount across all satial units in a time. In a more general sace-time setting, we consider changing the regressor by the same amount across all satial units in some consecutive time eriods, say, from the time t to t, where < t t t. We assume the situation that x s for t s t do not in uence W nt for t outside of the interval [t ; t ]. We can decomose the sace-time multilier into two comonents the rst comonent collects terms for the eriods from t + to t; and the second comonent for the eriods from t to t. 9

11 = = for t = t + ; :::; t, the rst comonent also vanishes. hus, we t = h=t t A (h) nt S n;t h l n + t (h) nt h=t S n;t h + n;t h (c n + X n;t h ), where l n is a n-dimensional vector of ones. Hence, the average total imact g t ( ; c n ) n nt) g t ( ; c n ) = " t t h=t t n lna (h) nt S n;t h l n + l (h) S n;t h A(h) n;t h is (c n + X n;t h ) hus, marginal changes in a dynamic model have imacts on future eriods. he A (h) nt S n;t h #. (9) rovides the sace-time multilier of X n;t h at time eriod t h on h-eriods ahead outcome Y nt. Here, A nt S n;t = S nt ( I n + W n;t )S n;t combines several satial and time transferring mechanisms. he rst comonent on the right hand side of (9) summaries the average total imact of changing x for all satial units in the consecutive eriods t to t on the exected outcomes Y nt at time t with W ns s xed. he second comonent catures those changing e ects due to the corresonding changes on satial weights (or network) matrices. For the secial case with t = t = t (so that we have the change of x only at the eriod t), the average total imact on the exected outcome E(Y nt ) will be simli ed to n l nsnt l n n l nt (c n + X nt ): () In the event that the satial weights matrix W nt does not deend on X nt, one has the average total imact as that in LeSage and ace (9) for a cross section SAR model (which is the rst term in ()), where S nt is regarded as the satial multilier matrix of X nt on the outcomes at t. When W nt deends on X nt in its construction, the second comonent n l n S nt on the (c n + X nt ) takes into account the marginal change of In addition, one may also consider the marginal change of x for a single individual at a articular time. he analysis can be easily modi ed with t = t, and that average total imact of an individual i with a change in x i;t will be nt ) n l n i;t n lna (t t) nt Sn;t e ni + ln where e ni is the ith unit vector of dimension n. (t nt + A i;t (t t) nt! (c n + X ), () i;t For the average total imact g t ( ; c n ) in (9), it is nonstochastic and deends on the true arameter value. If we relace those unknown arameters with estimates, we will have an estimate of the exectation and its variance can be obtained by the delta-method. roerties of g t (^ ; ^c n). he following theorem summarizes asymtotic

12 heorem 4 Under Assumtions is UB uniformly in t, it follows that, when n 3!, g t (^ ; ^c n) g t ( ; c n ) d! N(; lim! g; ), () where ^ is the bias corrected estimate in (6), ^c n ^c n (^ ) is from (7) and g; roof. See Aendix B.3. is de ned in (44). hus, the rate of convergence of g t (^ ; ^c n) is. Here, the ratio condition n! is needed because 3 we use the bias-corrected estimates in estimating g t ( ; c n ). As the weights matrices W nt are row-normalized, we have some interesting imlications of marginal e ects. Under row-normalization of W nt, n l ns nt l n = is a constant and we have n l ns nt l n = for the rst comonent in (), which is the satial multilier in a cross sectional SAR model. For the second comonent in (), by denoting x :t = n n i= x it as the cross sectional mean of x it at t, that comonent will deend on the deviation of x it from its cross sectional samle mean x :t : n l n his is so, because n l (c n + X nt ) = n l nx :t = [(c n l n c : ) + (X nt l n x :t ) ]. n l nsnt l n =, and, so is the term involving the mean ln c : of c n. hus, this comonent of imact would deend mainly on the variations of x it and c : across satial units. Here, this imact could also be zero if W nt is symmetric, i.e, a double stochastic matrix, because lns nt = ln is a constant vector in terms of column sums. In general, by regarding the above term as emirical covariance of l with c n+x nt, the e ect due to changes in W nt might be small if that emirical covariance is small. For the general case in Equation (9), the rst comonent (which would also be the average total imact for the case without x in W nt s) is t t h=t t n l na (h) nt S n;t h l n = t t h=t t + h. he second comonent of (9) will be t n A(h) nt S n;t h ) h=t t (c n c : l n + (X n;t h x :;t h l n ) ), which deends on c n and X nt deviated from their cross sectional means. he marginal e ects considered in revious aragrahs have the change of x, which aears in the SAR model additively as a direct exlanatory variable and may also aear in W nt s. In a model, some exlanatory variables might not aear as direct exlanatory variables in the outcome equation but aear in W nt s. Under this situation, the rst term of g t ( ; c n ) in (9) will be irrelevant. revious analysis can be roerly modi ed to accommodate such a relatively restricted imact.

13 5 Monte Carlo We conduct a Monte Carlo exeriment to evaluate the erformance of MLEs and the bias corrected estimator with time varying satial weights matrices. he DG is (): Y nt = W nt Y nt + Y n;t + W n;t Y n;t + X nt + c n + t l n + V nt ; t = ; ; :::;. We generate samles using a = (:; :; ; :5; ) and b = (:4; :; ; :; ), where = ( ; ; ; ; ), and X nt ; c n, t and V nt are generated from indeendent normal distributions. We use =, 5, and n = 49, 96. For time varying satial weights matrices, we choose an alternating attern. When t is odd, W nt is a square tessellation where each unit only interact with its left and right neighbors (for the left and right edge units, they have then only one neighbor). We call this a left-right matrix. When t is even, W nt is a queen matrix, which resents a square tessellation with a connectivity of eight for the inner elds on the chessboard and three and ve for the corner and border elds, resectively. his attern of time varying satial weights matrices is motivated by wet and dry seasons in Druska and Horrace (4), so that dry seasons imly a better tra c. All these weights matrices are row-normalized. For each set of generated samle observations, we obtain the MLE ^, construct the bias corrected estimator ^, and evaluate the biases ^ and ^. We do this 5 times. We also comare the emirical standard deviation (SD) and the emirical mean square error (RMSE) of these 5 estimates. Also, a coverage robability (C) is reorted. 3 With di erent values of n and, nite samle roerties of the estimators are summarized in ables -, where able is the result of estimates before bias correction and able is after bias correction. ables - here We can see that both estimators have some biases, and bias corrected estimators have smaller biases on average. he bias correction reduces those biases, which are originally larger. hese results are consistent with our asymtotic analysis in heorems and 3, because the bias corrected estimator will eliminate the bias of order O( ). We note that the bias reduction is achieved without signi cant increase in the variance of the bias corrected estimator. his results in smaller RMSEs for bias corrected estimates. For statistical inference, the Cs of the estimators before bias correction (under 95% con dence level) have lower values due to the bias, esecially when n is large (so that is relatively small) and esecially for Cs of and. After bias correction, the Cs are close to the seci ed 95% con dence level. For di erent cases of n and, when is larger, biases and SDs are smaller. Also, for each given, when n becomes larger, the biases remain nearly the same, but variances will be smaller. his is consistent with

14 our theoretical results, as the bias is of the order O( ) and variances of the estimators are of the order O( ). For di erent values of, there seems no aarent change with. We reort some results using misseci ed time invariant weights matrices, while the DG still has the above alternating attern. ables 3 and 4 are cases using the time average of left-right and queen matrices. ables 5 and 6 are cases using the left-right matrix only, and ables 7 and 8 use the queen matrix only. Also, ables 9 and are cases using the rook matrix 4 only, which is another intermediate way of time invariant aroximation instead of time average. ables 3- here When we use the time average of the left-right and queen matrices, from ables 3-4, the erformance of estimates under this misseci cation is not as good as the correct seci cation in ables -. Esecially, after bias correction, this misseci cation will yield large bias of for the DGs with a larger. For the DG with a smaller, the misseci cation of time average still yields accetable results. From ables 5-6, when we use the misseci cation of a left-right matrix, we have large biases in, and even after bias correction, esecially for the DG with a larger. From ables 7-8, when we use the misseci cation of a queen matrix, we have large biases in when is small, and large bias in,, and when is large. From ables 5-8, by comaring the left-right matrix (underseci cation) and the queen matrix (overseci cation), we see that overseci cation has a better result than the underseci cation in the estimation and inference of. Finally, instead of time average, we use the misseci cation of rook matrix only, which is an intermediate case of left-right and queen matrices. From ables 9-, we see that the rook matrix misseci cation yields large bias for and, esecially when is large. For the DGs in ables -, with alternating left-right and queen matrices, the satial weights matrices are varying substantially over time. We also conduct a small Monte Carlo to investigate the erformance of estimates with slowly time varying weights matrices in the DG. We choose alternating rook and queen matrices as the true time varying satial weights matrices, and the results are reorted in ables and. We see that estimates with the correct seci cation have small bias and adequate Cs. However, the misseci cation of the time average, rook only and queen only matrices, would be of less concern because the bias is not too large and Cs are adequate for most cases, 5 comared to ables 3-8. Also, the underseci cation yields larger bias on average than the overseci cation of the satial weights matrices, esecially for the estimate of the satial e ect. ables - here 3

15 In the following ables 3-4, we investigate the marginal e ect with the receding W nt in the DG. As these W nt s are row-normalized and do not deend on exogenous variables X nt, the marginal e ect for one eriod, from (), is. With estimates for n = 49, = and a from ables -, we lot these estimates of with 5 reetitions. able 3 is before the bias correction, and able 4 is after the bias correction. Under the correct seci cation of time varying satial weights matrices, the estimate of the marginal e ect 6 is centered around the true value = :5 =, and its emirical standard deviation is :48. Under the misseci cation with time average of the weights matrices, the estimate of is also around the true value, but its emirical standard deviation is larger (which is :6). Under the misseci cation with a left-right matrix only, the estimated marginal e ect is.64, which under-estimates the true value by %. If we estimate the marginal e ect under the misseci cation with a queen matrix only, it is around.8, which has under-estimated the true value by %; but, it has a much larger variance than the left-right matrix case (:8 vs. :7). Finally, for the misseci cation of a rook matrix only, it is also under-estimated around.8, with a standard deviation :7 between the cases of left-right only and queen only matrices. 6 Conclusion ables 3-4 here his aer investigates the SDD model with time varying satial weights matrices. he QML estimate is consistent and asymtotically normal under this setting. As we see in the Monte Carlo, a substantial model misseci cation with a time invariant satial weights matrix, where the true rocess has time varying satial weights matrices, will cause biases in estimates. Also, time varying satial weights matrices have imlications on the sace-time multilier and marginal e ects of regressors. When using economic distances to construct the satial weights matrix, we might have a searate issue of endogeneity as these economic distances could be endogenous. his calls for the estimation of satial models with endogenous satial weights matrix, which is beyond the scoe of this aer and is an interesting toic for future research. Notes Matlab code is available on request. We would like to emhasize that the time varying W nt is due to changes of economic environments, but not due to missing observations. Missing observations would create an observable unbalanced anel, which is beyond the scoe of this aer. 3 As W n is row-normalized, whether elements of D n are larger than or not deends on the value of + +. See Yu et al. (7). 4

16 4 When W nt is row-normalized for all t, unit roots will resent when + + =. his is so, because + is the eigenvalue of A nt with corresonding eigenvector of ones l n for all t when W nt is row-normalized (so that A ntl n = + l n), where we have W ntl n = l n for all t. Hence, when + =, we have at least one unit root in A nt for all t. 5 A formal condition which guarantees the in nite sums being well de ned is Assumtion 6 below. 6 his follows from row-normalized W nt, which imlies that S n;t h ln = l n for all h and A (h) nt ln = + h ln. 7 For asymtotic analysis, we assume that both n and are large. If n is nite, the model with satial interaction with nite n would be of less interest. If is nite, the MLE would have an incidental arameter roblem and alternative estimates such as the GMM shall be designed. 8 We say a (sequence of n n) matrix B n is uniformly bounded in row and column sums if su n kb nk < and su n kb nk <, where kb nk su n in j= jb ij;nj is the row sum norm and kb nk = su n jn i= jb ij;n j is the column sum norm. q q q 9 (h ) When W nt is time invariant, A nt becomes A h n and a ;n in (4) can be written as n a ;n = n ' n ()+O( n 3 ) n tr J n h= Ah n () Sn () n where ' n () = tr J nw n h= Ah n () Sn () B n tr(jngn() h= Ah n () Sn ()) + n tr(jngn()wn h= Ah n () Sn ()) + n tr(jngn()) q q hus, for the time invariant W n, as is shown in Lee and Yu (c), we have = n ' n ( ) + O( n 3 ) + O ( ) q where the O( n 3 ) term comes from small terms in (4). In LeSage and ace (9), for a cross sectional SAR model Y n = W ny n + X n + V n with S n = I n W n, the average total imact is n l n S n l n, which can be decomosed into the average direct imact n tr(s n ) and the average indirect imact n (l ns n l n tr(s n )). If we have changes of x for all time from the in nite ast, such a total imact will be ( + + ). We generate the data with + eriods and take the last eriods as our samle. And the initial value is generated as N(; I n) in the simulation. 3 For the estimation of the information matrix, we use (n ) t= ( Z ~ nt; W^ nty nt) J n( Z ~ nt; W^ nty nt) to aroximate! EH c + (kx+)(kx+) (kx+). (kx+) (n ) t= tr((jngnt) ) 4 he 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. 5 For the misseci cation with the rook matrix only, the C is very small when is large. 6 he following statistics reorted are all after bias correction.. C A Aendices A Lemmas and Algebra A. Some Lemmas Denote U nt = h= nt;hv n;t+ h and W nt = h= Q nt;hv n;t+ h, where nt;h and Q nt;h are n n nonstochastic matrices. Let U ~ nt = U nt U where U = t= U nt =, and U n;t = U n;t U; where U ; = t= U nt =. Also, Wnt ~, W and V ~ nt are de ned similarly. Assumtion A. he disturbances fv it g, i = ; ; :::; n and t = ; ; :::; ; are i:i:d across i and t with zero mean, variance and E jv it j 4+ < for some >. 5

17 Assumtion A. nt;h = A nt (h) nt B n;t h for h = ; ; ::: with (h) nt = nt n;t n;t h+ and nt; = I n for all t, where A nt and B n;t h are UB invertible square matrices uniformly in t. Similarly, Q nt;h = C nt Q (h) nt D n;t h, where C nt and D n;t h are UB invertible square matrices. Furthermore, su t k nt;h k h= and su t k nt;h k are uniformly bounded for all n where kk and kk are row and column sum norms. h= Similar UB conditions hold for Q nt;h s. Assumtion A3. he elements of n vector D nt are nonstochastic and bounded, uniformly in n and t. Assumtion A4. n is a strictly increasing function of, and goes to in nity. Assumtions A.-A.4 are needed for the following lemmas. hese assumtions can all be imlied from assumtions in the main text. In articular, the stochastic comonent U nt = h= A(h) nt S n;t h V n;t h of Y nt in (3) is a secial form of U nt = h= nt;hv n;t+ h with nt;h = A nt nt n;t n;t h+ B n;t h in Assumtion A., where A nt = I n, nt = A nt and B n;t h = S n;t h. Also, for the stochastic comonent of W nt Y nt, which is W nt h= A(h) nt S n;t h V n;t h, it has A nt = W nt, nt = A nt and B n;t h = S n;t for the following lemmas are available on request. Lemma Under Assumtions A, A and A4, and h. roofs t= U ~ ~ nt W nt E( t= U ~ ~ nt W nt ) = O, (3) U n n; V E U n n; V = O, (4) t= V ~ nt ^B nt V nt = ( ) where E( ~ t= U ~ nt W nt ) = O() and E nu n; V = tr t= tr(b nt) + O, (5) t t= h= nt;h. With Z nt = (Y n;t ; W n Y n;t ; X nt ), we rovide some lemmas related to ~ Z nt, Z and ~ V nt, V. Denote ^ B nt Z nt = B nt Z nt t= B ntz nt and B Z = t= B ntz nt. Lemma Under Assumtions -7, for any n n nonstochastic UB matrices B ;nt and B ;nt, and where E t= X B^ ;nt Z nt B^ ;nt Z nt E t= X B^ ;nt Z ~Vnt nt E t= B^ ;nt Z nt B^ ;nt Z nt is O() and E X B^ ;nt Z nt B^ ;nt Z nt = O ; (6) t= X B^ ;nt Z ~Vnt nt = O ; (7) t= t= 6 B^ ;nt Z ~Vnt nt is O.

18 Lemma 3 Under Assumtions -7, for any n n nonstochastic UB matrices B ;nt and B ;nt, and n B ; Z B; Z n B ; Z V n B ; V V E n B ; Z B; Z = O E n B ; Z V = O E n B ; V V = O, (8), (9), (3) where E n B ; Z B; Z is O(), E n B ; Z V is O and E n B ; V V is O. From (), ~ Z nt = ~ Z (u) nt ( U ; ; W n; U ; ; nkx ), where ~ Z (u) nt = [(~ n;t c n + ~ X n;t +U n;t ); ( W n;t ^ n;t c n + W n;t ^ X n;t + W n;t U n;t ); Xnt ~ ] with X ~ n;t = X n;t t= X n;t. Hence, ~Z nt has two comonents: one is Z ~ (u) nt, uncorrelated with V nt ; the other is ( U ; ; W n; U ; ; nkx ), ^ correlated with V nt when t. Denote B j;nt Z (u) nt = (( B j;nt ^ n;t c n + B j;nt ^X n;t + B j;nt U n;t ); ( B j;nt W^ n;t n;t c n + B j;nt W^ n;t X n;t + B j;nt W n;t U n;t ); B^ j;nt X nt ) for j = ;. Following is a lemma related to ~ Z (u) nt and Z nt. Lemma 4 Under Assumtions -7, for any n n nonstochastic UB matrices B ;nt and B ;nt, E X t= B^ ;nt Z nt B^ ;nt Z nt E X t= ^ B ;nt Z (u) ^ nt B ;nt Z (u) nt = O, where E t= ^ B ;nt Z (u) ^ B ;nt Z (u) nt nt is O(). For the central limit theorem we use in this aer, it has the following form: Q = X t= U n;t V nt + D ntv nt + V ntb nt V nt trb nt ; where B nt is an arbitrary n n symmetric UB matrix (the assumtion that B nt is symmetric is maintained w.l.o.g. since V ntb nt V nt = V nt[(b nt + B nt)=]v nt and D nt is an n-dimensional nonstochastic vector. hen, the mean of Q is Q Q = 4 tr t= h= = and its variance is X X X X nt;h nt;h!+ DntD nt + t= t=! X n X nx b nt;ii + 4 tr(bnt) + 3 d nti b nt;ii ; where s = Ev s it for s = 3; 4, b nt;ii s are diagonal elements of B nt and d nti is the ith element of D nt. Lemma 5 Under Assumtions A-A4 and that B nt is UB uniformly in t, if the sequence Q Q away from zero, then, d Q! N(; ). i= t= i= is bounded 7

19 A. First and Second Order Derivatives of (7) Using trg n () tr(j n G n ()) = and tr(g n()) tr((j n G n ()) ) = ( ) (see Lee and Yu (c)), the rst order derivatives of (7) ln L n; = and the second order derivatives are ~ t= ZntJ n Vnt ~ () ( ^W nt Y nt ) J n Vnt ~ () C tr(j n G nt ()) A, (3) t= 4 t= ( ~ V nt()j n ~ Vnt () (n ) ) ln L n; (3) B t= Z ~ ntj n ~ Znt t= ( ^W nt Y nt ) J n Znt ~ t= ( ^W nt Y nt ) J n ^W nt Y nt + tr((j n G nt ()) ) C A ~ 4 t= Vnt()J n Znt ~ ~ 4 t= Vnt()J n ^W nt Y nt (kx+)(k x+) ~ (kx+) B 4 t= ZntJ n Vnt ~ (kx+) 4 t= ( ^W nt Y nt ) J n Vnt ~ () (n ) (kx+) + ~ 4 6 t= Vnt()J n Vnt ~ () C A. Hence, with ^W nt Y nt = W nt Y nt t= W nty nt and W nt Y nt = G nt Z nt + G nt c n + G nt l n t + G nt V nt where G nt l n t = t l n due to row-normalization of W nt, we have ^W nt Y nt = ^ G nt Z nt + ~ G nt c n + ~ t l n + ^G nt V nt. hus, the score is (8) because J n l n =. For the information matrix, denoting H c = (n ) t= ( Z ~ nt ; ( G^ nt Z nt + G ~ nt c n )) J n ( Z ~ nt ; ( G^ nt Z nt ~G nt c n )), the information matrix is equal to ; = E ln L n; ( ) (n = (kx+)(k EH c x+) B (kx+) (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) C (kx+3) A (kx+) (n ) t= tr(j ng nt ) 4 (kx+)(k x+) (n )E(G V ) J nz 4 (n )E V J nz (n )E (G Z ) J n G V + (n ) tr( t= G ntj n G nt ) 4 (n )E (G Z ) J nv + (n ) tr( t= J ng nt ) 4 where the second term is O from Lemma 3 with G V = t= G ntv nt and G Z = t= G ntz nt. A.3 Nonsingularity of the Information Matrix We obtain the nonsingularity by an argument by contradiction (similar to Lee (4)). For lim! ;, we need to rove that = imlies = where = ( ; ; 3 ), ; 3 are scalars and is (k x +) vector. If this is true, then, columns of would be linear indeendent and would be C A, 8

20 nonsingular. Denote H = lim ~! (n ) t= ZntJ n Znt ~, H = lim! ~G nt c n ), H = H and H = lim! (n ) ~ (n ) t= ZntJ n ( G^ nt Z nt + t= ( G^ nt Z nt + G ~ nt c n ) J n ( G^ nt Z nt + G ~ nt c n ). We have = EH EH (kx+) EH EH + lim! (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) C A (kx+) lim! (n ) t= tr(j ng nt ) + (kx+)(k x+) (kx+) (kx+) (kx+) lim! (n ) t= tr(j ng nt ) C A. (kx+) hus, = imlies and 8 >< >: EH + EH + lim! (n ) t= + lim! (n ) lim! (n ) EH + EH = ; tr(g nt J n G nt ) + tr((j n G nt ) ) 9 >= = : tr(j n G nt ) 3 >; t= t= tr(j n G nt ) + 3 = : he rst equation imlies = (EH ) EH and the third imlies 3 = lim! (n ) t= tr(j ng nt ). By eliminating and 3, the second equation becomes 8 < EH EH (EH ) EH + : lim! (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) h lim! t= 9 = i tr(j ng nt) ; = : n herefore, the nonsingularity of the information matrix requires that (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) h i tr(j ng nt) t= n > or EH c tr(j is nonsingular. For each t, denoting C nt = J n G ng nt) nt n I n, we have tr(g ntj n G nt ) + tr((j n G nt ) ) tr (J n G nt ) n his imlies (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) t= = tr [(C nt + C nt )(C nt + C nt ) ] : h i tr(jng nt) n. From the Cauchy inequality (so that (a +a + +a ) (a +a + +a ) for any scalar a t ), we have t= h i tr(j. ng nt) t= n By combining the above, we have X tr(g (n ) nt J n G nt ) + tr((j n G nt ) ) X t= t= tr(j n G nt ) : n h i tr(jng nt) n 9

21 A.4 Variance Matrix of the Score For the variance matrix of the score, as ~ Z (u) nt E( (n ln L (u) n; ln (u) n; () (n is uncorrelated with V nt, we have ) = A ; + A ; where (n ) E ~ t= Z (u) nt J n ~ (u) Z nt A ; = (n ) E t= ( G ^ nt Z (u) nt + G ~ nt c n ) J n Z ~ (u) C nt A (kx+) (kx+)(k x+) B (kx+) (n ) E t= ( G ^ nt Z (u) nt + G ~ nt c n ) J n ( G ^ nt Z (u) nt + G ~ C nt c n ) A (kx+) (kx+)(k x+) B (kx+) (n ) t= tr(g nt J n G nt ) + tr((j n G nt ) ) C A, (kx+) (n ) t= tr(j ng nt ) 4 and (kx+)(k x+) B 3 n A ; 4 (n ) t= i= (J ng nt ) ii E(J n Z ~ (u) nt ) i C A 3 n 6 t= i= E(J ~ n Z (u) nt ) i (kx+)(k x+) + B 3 n 4 (n ) t= i= (J ^ ng nt ) ii E(J ngnt Z (u) nt + J n Gnt ~ c n ) i C 3 A n (kx+) 6 t= i= E(J ^ ng nt Z (u) nt + J n Gnt ~ c n ) i (kx+)(k x+) B n (kx+) 4 (n ) t= i= (J ng nt ) ii C A (kx+) 6 (n ) t= tr(j ng nt ) For the rst matrix A ;, it is equal to ; + O using Lemma 4. For the second matrix A;, as ~ t= Z ln L nt =, we have E( (u) n; ln L (n (u) n; () (n ) = ; + ; + O where ; = 3 4 (n ) t= i= 3 6 t= (kx+)(k x+) n (J n G nt ) ii E(J n Z ~ (u) nt ) i n i= E(J ~ n Z (u) nt ) i (kx+)(k x+) 3 n ^ (kx+) B 4 (n ) (J n G nt ) ii E(J ngnt Z (u) nt + J n Gnt ~ c n ) i t= i= n + + A (n ) (J n G nt ) ii t= i= 3 n B 6 t= i= E(J ^ ng nt Z (u) nt + J n Gnt ~ c n ) i C A (n ) tr(j n G nt ) t= C A (33) : C A

22 When V nt are normally distributed, ; = (kx+4)(k x+4) because 3 = and 4 with a time invariant W n, 3 would not lay a role in ; as shown in Yu et al. (8)). 3 4 = (for the case B roof for heorems B. roof for heorem From Y nt = W nt Y nt + Z nt + c n + t l n + V nt, we have Y nt = S nt (Z nt + c n + t l n + V nt ). As V nt () = S nt ()Y nt Z nt c n t l n and S nt ()S nt = I n ( )G nt, we have V nt () = S nt ()S nt (Z nt + c n + t l n + V nt ) Z nt c n t l n = ( )G nt (Z nt + c n + t l n ) + Z nt ( ) + V nt +( )G nt V nt + c n c n + ( t t )l n. hus, J n Vnt ~ () = ( )J n ( G^ nt Z nt + G ~ nt c n ) + J n Znt ~ ( ) + J n Vnt ~ + ( )J n ^G nt V nt because J n l n =. Hence, with H c = (n ) t= ( Z ~ nt ; ( G^ nt Z nt + G ~ nt c n )) J n ( Z ~ nt ; ( G^ nt Z nt + G ~ nt c n )), we have t= V (n ) ~ nt()j n Vnt ~ () (34) = ( ; )H c ( ; ) + t= (n ) ( V ~ nt + ( ) ^G nt V nt ) J n ( V ~ nt + ( ) ^G nt V nt ) + t= (n ) (( )( ^ G nt Z nt + ~ G nt c n ) + ~ Z nt ( )) J n ( ~ V nt + ( ) ^G nt V nt ). In the following, we show (i) the uniform convergence of (n ) ln L n; () (n ) E ln L n; () to zero in robability uniformly in in ; () the uniform equicontinuity of (n ) E ln L n; (), and (3) global identi cation. he consistency of the QMLE ^ then follows. o rove (n ) ln L n; () From (34), by Lemma, we have (n ) (n ) E ln L n; ()! uniformly in in any comact arameter sace : ~ t= Vnt()J n Vnt ~ () (n ) E ~ t= Vnt()J n Vnt ~ ()! uni- (n ) (n ) formly in in, because and are bounded in. As ln L n; () = ln ln + t= ln js nt()j ln( ) ~ t= Vnt()J n Vnt ~ (), using the fact that is bounded away from zero in, we have = (n ) ln L n; () ~V (n ) nt()j n Vnt ~ () t= (n ) E ln L n; () (n ) E t= ~V nt()j n Vnt ~ ()! uniformly in.

23 o rove (n ) E ln L n; () is uniformly equicontinuous in in any comact arameter sace : We have E (n ) ln L n; () = ln ln + (n ) (n ) E X t= X ln js nt ()j t= ~V nt()j n ~ Vnt (): ln( ) n From (34), by (5) in Lemma, we have (n ) E t= ( V ~ nt + ( ) ^G nt V nt ) J n ( V ~ nt + ( ) ^G nt V nt ) = + ( ) t= (n ) tr(g ntj n G nt ) + ( ) = = t= + ( ) t= (S nt()s nt ) J n (S nt ()S = t= nt() + O, where nt() is in (). Also, from Lemma, (n ) t= tr(j ng nt ) n tr(g ntj n G nt ) + ( ) n tr(j ng nt ) + O nt ) + O (n ) E t= (( )( ^ G nt Z nt + ~ G nt c n ) + ~ Z nt ( )) J n ( ~ V nt + ( ) ^G nt V nt ) = O ( ), uniformly in in because it is a olynomial function in and is a bounded set. hus, E (n ) X t= ~V nt()j n ~ Vnt () = (n ) ( ; )EH c ( ; ) + + O t= nt()+o. (35) he rst term is a quadratic form in. he second term has nt() = n (S nt()s nt ) J n (S nt ()S nt ), which are all olynomial functions of. hus, E (n ) ln L n; () is uniformly equicontinuous in. o rove global identi cation: As E ~ t= VntJ n Vnt ~ = (n )( ), at, we have E ln L n; ( ) = t= ln js (n )( ) ntj ln( ). From (35), it follows that (n ) E ln L n; () (n ) E ln L n; ( ) t= ln js nt()j = (ln ln ) + (n ) (n ) t= E ~ V nt()j n ~ Vnt () = ;n (; ) ;n; (; ) + o() (n ) ln (n ) ln + t= ln js ntj + n ln( ) n ln( )

24 where ;n (; ) = (ln ln ) + t= ln js nt()j + n ln( ) n ln( ) ( t= ln js ntj t= nt() ) and ;n; (; ) = (n ) ( ; )EH c ( ; ) : Consider the rocess Y nt = W nt Y nt + t l n + V nt for each eriod t, the log likelihood function of this rocess after the elimination of t is ln L ;n (; ) = n ln n ln + ln js nt ()j ln( ) (S nt ()Y nt ) J n S nt ()Y nt. Let E () be the exectation oerator for Y nt based on this rocess. It follows that E ( n ln L ;n(; )) E ( n ln L ;n( ; )) = (ln ln )+ n ln js nt()j n ln js nt( )j+ n ln( ) n ln( ) ( nt() ). By averaging these t = ; :::; eriods, we have ;n (; ). By the information inequality, ln L ;n (; ) ln L ;n ( ; ). hus, ;n (; ) for any (; ). Also, ;n; (; ) is a quadratic function of and. Under the condition that lim! EH c is nonsingular, ;n; (; ) > whenever (; ) 6= ( ; ), so, (; ) is globally identi ed. Given, is the unique maximizer of ;n ( ; ). Hence, (; ; ) is globally identi ed. When lim! EH c is singular, and cannot be identi ed from ;n; (; ). Global identi cation requires that the limit of ;n (; ) is strictly less than zero. As ;n (; ) by the information inequality, lim! ;n (; ) 6= is equivalent to lim! (n ) t= ln Snt Snt 6= lim! (n ) t= ln nt ()Snt ()Snt (). Here, the term n ln( ) n ln( ) in ;n (; ) will disaear when n becomes larger. After and are identi ed, given, can be identi ed from ;n; (; ). B. roof for heorems and 3 For the bias term, we can aly Lemma. For the comonent n Y n; J n V, we have Similarly, E Y n n; J nv h = (n ) tr J n t= E n W n; Y ; JnV h = (n ) tr J n t= t ) h= A(h n;t+h t h= W n;t+h A i Snt. (h ) n;t+h i Snt, 3