Panel VAR Models with Spatial Dependence

Transcription

1 Panel VAR Models with Spatial Dependence Jan Mutl y Institute of Advanced Studies February 2, 29 Abstract I consider a panel vector-autoregressive model with cross-sectional dependence of the disturbances characterized by a spatial autoregressive process. I propose a threestep estimation procedure. Its rst step is an instrumental variable estimation that ignores the spatial correlation. In the second step, the estimated disturbances are used in a multivariate spatial generalized moments estimation to infer the degree of spatial correlation. The nal step of the procedure uses transformed data and applies standard techniques for estimation of panel vector-autoregressive models. I compare the small-sample performance of various estimation strategies in a Monte Carlo study. Keywords: spatial PVAR; multivariate dynamic panel data model; spatial GM; spatial Cochrane-Orcutt transformation; constrained maximum likelihood estimation JEL Codes: C3, C3, C33 Introduction Vector autoregressive (VAR) models are extensively used in econometric applications in a wide variety of elds. The extension to panel data represents an interesting challenge due to the likely presence of cross-sectional heterogeneity. In this paper I tackle the issue by considering a panel VAR model with a particular class of dependence structure in the disturbances. I consider the situation where the time dimension T is xed. As a result, the correlations across cross-sectional units have to be parsimoniously parameterized in order to avoid the incidental parameters problem. I follow the spatial econometrics literature and study a rst order spatial autocorrelation model with a known spatial weighting matrix. Of course there are other ways to specify the spatial dependence in the model and prominent I am grateful to Michael Binder, Ingmar Prucha, Harry Kelejian and Robert Kunst for helpful advice and comments. However, all errors and omissions are my responsibility. y Stumpergasse 56, A-6 Vienna, Austria, tel , fax , mutl@ihs.ac.at. This would be, for example, the case in the seemingly unrelated regressions model with xed time dimension. There, the number of parameters grows quadratically with the sample size.

2 alternatives include a non-parametric speci cation generalizing Conley (99) and Chen and Conley (2), or common factor speci cation inspired by work of Spearman (94); see e.g. Mulaik, 972, or Ng and Bai, 28, for a more recent treatment and overview. The panel spatial autocorrelation model considered in this paper is a generalization of spatial econometric models that include single equation models, e.g., Cli and Ord (973, 98), and simultaneous equation models, such as Whittle (954), Anselin (988) or Kelejian and Prucha (998, 999 and 24). Lee (24) provides formal large sample results for a (quasi) maximum likelihood estimator of a static single cross-section model. Extensions to panel data with single equation include Lee and Yu (28) who extend the formal results for the maximum likelihood estimation for static panel data models, and Kapoor et al. (27) who introduce and derive formal large sample results for the spatial generalized moments method. On the other hand, the current paper extends the panel VAR literature to allow for cross-sectional dependence of the model disturbances; for models with independent disturbances see, e.g., Binder et al. (25) for the quasi maximum likelihood (QML) and minimum distance (MD) estimators, or Arellano and Bond (99), Ahn and Schmidt (995) and Arellano and Bover (995) for the generalized method of moments (GMM) approach in a single equation framework. The next section will specify the model and state the assumptions maintained throughout the paper. Section 3 describes the various estimation procedures, while Section 4 presents the results from a Monte Carlo study comparing small-sample performance of these estimators. Section 5 then concludes. 2 The Panel VAR Model In this section I specify the model and discuss the main assumptions that will be maintained throughout. The speci cation adopted here uses the spatial autoregressive framework with known spatial weighting matrix to capture the heteroscedasticity in the data. Hence it replaces the assumption that the disturbances of the model are independent among units and as such can be viewed as an alternative to other approaches such a principle component models. The model under consideration can be expressed as a rst order panel VAR model: y it y i;t + u it ; () u it w ij u jt + (I m ) i + " it j where the rst subscript i 2 f; ::; Ng refers to the cross-sectional dimension and the second subscript t 2 f; ::; T g refers to the time dimension of the panel of observations fy it g in tt. I also allow the model to contain more than one equation and so the observations y it, the individual-speci c e ects i and the disturbances u it and " it are m vectors and the known weighting parameters w ij, the unknown model parameters and the identity matrix I m are all m m matrices. The degree of spatial autocorrelation is captured by the scalar 2

3 parameter. Note that I restrict the individual e ects to be of the form (I m ) i so that when the model contains units roots (for example when I m ), the trending behavior remains the same as in the stationary case. Stacking across individuals we obtain where y t (I N ) y t + u t ; (2) u t Wu t + [I N (I m )] + " t ; y t (y t; ::; y Nt) ; mn u t (u t; ::; u Nt) ; mn ( ; ::; N) mn " t (" t; ::; " Nt) mn (3) and the mn mn weighting matrix W is w w N B w N w NN Solving for the disturbance terms yields 2 C A mnmn (4) u t (I mn W) ([I N (I m )] + " t ) : (5) Observe that the solution to the disturbance process depends on the sample size N and hence the disturbances as well as the observed process y it form triangular arrays. However, I omit the indexation by N in order to maintain legible notation. I will next motivate and then formulate the basic set of maintained assumptions. These will consist of an assumption on the innovations " it (e.g. that they are independently distributed), an optional assumption on the individual e ects i, restrictions on the spatial weights W and the parameter space () that guarantee stability in the spatial dimension, and naly an assumption on how the initial observation of the process was generated. 2. Random vs. Fixed E ects Speci cation Allowing for individual e ects without any additional restrictions, leads to an incidental parameters problem. As the time dimension of the panel is xed, one cannot consistently estimate a general form of the individual-speci c e ects with a nite number of observations per parameter. To resolve this problem, there are two options. Either to assume that there is a well-behaved distribution (e.g. with nite fourth moments) from which the individualspeci c e ects are generated (the random e ects speci cation), or transform the data to obtain speci cation that does not contain the individual-speci c e ects (the xed e ect speci cation). The usual approach in the xed e ect speci cation is to rst-di erence the 2 Note that this assumes that the inverse of (I mn W) exists. This will indeed be the case under the assumptions maintained in the paper (see Section 2.3). 3

4 data; see the argument in Hsiao, Pesaran and Tahmiscioglu (22) who show in a univariate context that the QML estimator is invariant to the choice of the transformation matrix that eliminates the individual-speci c e ects. The argument is readily extended to the multivariate setting in this paper. However, the xed e ect speci cation and rst-di erencing does not eliminate the incidental parameter problem unless we assume that the spatial weighting matrices are constant over time. Hence the choice between xed and random e ects speci - cation depends on which of the two assumptions (constant weighting matrix or existence of the distribution that generates the individual-speci c e ects) is more appropriate. In this paper I use the transformed likelihood approach (e.g. xed e ects speci cation). Nevertheless, the initial estimation procedure I suggest in this paper, can incorporate the random e ects assumption. In particular, in the second step of the procedure is a spatial generalized moments method that uses estimated disturbances from the rst step. The spatial GM estimation provides estimates of the degree of spatial autocorrelation in the disturbances, as well as an estimates of the variance covariance matrices of both the independent innovations and the individual e ects. 3 Hence an extension of the likelihood approach to include individual e ects would be straightforward and it is easily implemented using the procedure discussed in this paper. 2.2 Initial Disturbances Speci cation Instead of conditioning on initial observations, I explicitly treat the initial conditions when de ning the likelihood function. There are several assumptions one can make. Since the data is not observed beyond the time period, the initial observations y are just equal to the initial disturbances (which now potentially include all the lagged e ects), i.e. I assume that u is spatially correlated and is generated by y u : (6) u Wu + + ; (7) where ( ; ::; N) with each i being an m vector of independently (of j and " jt, t > ) and identically distributed initial random disturbances with a constant (over i) variance covariance matrix. Hence the initial observations in rst di erences are y (I N ) y + u y (8) u [I N (I m )] u (I mn W) (" [I N (I m )] ) : I denote by the variance covariance matrix of the initial observation in rst di erences 3 However, the unweighted spatial GM procedure does not utilize the random e ects assumption and hence is suitable for the xed e ects speci cation. 4

5 (y ) after the spatial autocorrelation is removed, i.e. V C [(I mn W) y ] ; (9) where the notation V C (:) stands for variance covariance matrix. Given that 6 I m, we have that " + (I m ) (I m ) ; () and hence is unconstrained and the entries in it will enter as additional parameters into the likelihood function. In the pure unit root case ( I m ), the variance of the initial innovations is constrained to be ". In general, if the eigenvalues of are inside the unit circle, one could make further assumptions on the disturbances and express in terms of and ". In particular, since in this case the data generating process is dynamically stable and, therefore, one could assume that it has started in an in nite past. This would imply that the initial observations y are drawn from the limiting stationary distribution of the process, e.g. that: X y (I mn W) (I N ) j [I N (I m )] + " j j () Therefore, the initial observations in rst di erences are X y (I mn W) (I N ) j " j (2) j and! X V C [(I mn W) y] I N V C j " i; j j I N V C " i + (I m ) " I N " + (I m )! X j " i; j j! # X j " j (I m ) ; j (3) or given the de nition of, we have " " + (I m )! # X j " j (I m ) : (4) j Hence, I distinguish three assumptions on how the elements of. (UR) In the pure unit root case ( I m ), we have to set ". are determined: 2. (IOR) When all of the eigenvalues of are inside the unit circle, we could impose an additional assumption and restrict the elements of to be a function of " and, 5

6 i.e. or rewriting the expression in the equation above: vec Dvech " + [(I m ) (I m )] [I ( )] Dvech " ; (5) where D is a duplication matrix such that vec " Dvech ". 3. (NR) No restrictions are placed on elements of (other then imposing that is symmetric and strictly positive de nite matrix). In all of the cases, we have that the variance covariance matrix of the rst di erence of the initial observations is 2.3 Maintained Assumptions E (y y ) (I mn W) (I mn W ) : (6) In order to guarantee that the model and the estimation procedure is well de ned, I maintain the following assumptions about the disturbances and the spatial weighting matrices. Assumption The disturbance vectors " it are identically and independently (of " js for j 6 i) distributed with zero mean, and nite absolute 4 + moments for some >. Furthermore, the vector " it has a nite positive-de nite variance matrix ". The above assumption is needed to ensure that the observable data, which is a transformation of the " it process, has a well-de ned asymptotic properties. The next two assumptions ensure that the weighting matrices do not explode as the sample size increases. Assumption 2 The matrices (I mn W) are nonsingular for all jj < (W), where (:) denotes the spectral radius of a matrix. Furthermore, the parameter also satis es jj < (W). W) are uni- Assumption 3 The row and column sums of the matrices W and (I mn formly bounded in absolute value. Finally, I formalize the discussion in the preceding section into an assumption on the generation of the initial observation in rst di erences: Assumption 4 The initial observations y i are identically and independently (of " jt for t > ) distributed with zero mean, and nite absolute 4 + moments for some >. Furthermore, the vector y i has a nite positive-de nite variance matrix, given by one of the following: (UR) ", and I m, (IOR) vec Dvech " + [(I m ) (I m )] [I ( )] Dvech ", and 6 I m, (NR) no restrictions are placed on, and 6 I m,. 6

7 3 Estimation The model can be estimated using a variety of approaches. Straightforward least squares estimation of the rst di erences of the observations on its lagged values is not consistent because the error term u t is correlated with the explanatory variable y t. However, based on results in Mutl (26), there is an alternative instrumental variable (IV) estimation that leads to a consistent estimates of the spatially correlated disturbances. Given an initial estimator of the slope coe cients, we can then use a spatial generalized moments estimation (spatial GM) to obtain a consistent estimator of the spatial parameter ; e.g. use the moment conditions based on the estimated disturbances: bu t y t I N b IV y t (7) where b IV is the IV estimator of. Generalizing the univariate results in Kapoor et al. (27), it then follows that this two stage procedure leads to a consistent estimator of. Finally, in the last step of the proposed estimation procedure, we can use the spatial Cochrane-Orcutt transformation and write the model as 4 (I mn W) y t (I mn W) (I N ) y t + " t : (8) If is known, the transformed model can be estimated with standard techniques, such as the quasi-maximum likelihood (QML) method in Binder, et al. (25) or a multivariate extension of the generalized method of moments (GMM) approach of Arellano and Bond (99), Ahn and Schmidt (995), or Arellano and Bover (995). An alternative to the above procedure is to use the maximum likelihood function of the entire model and obtain a QML estimate. Given the computational complexity of this approach, is important to have reasonable initial estimates. Hence even if one ultimately employs the full likelihood approach, it is of interest to study the properties of the initial estimators. In the following, I rst de ne the IV estimator and then discuss the spatial GM estimator of the spatial parameter. Finally, I de ne the full as well as the constrained QML procedures. 3. Initial Estimation Unlike the transformed likelihood approach, the initial estimators are based on moment conditions that involve (lagged) levels of the endogenous variable. Therefore, their large (as well as small) sample properties are not independent of the distribution of the individual e ects. Similarly, the spatial GM procedure is based on estimated levels of the disturbances and directly uses a random e ects assumption. Hence I maintain the following assumption: Assumption 5 The disturbance vectors i are identically and independently (of " jt, j, and y j ) distributed with zero mean, and nite absolute 4 + moments for some >. Furthermore, the vector it has a nite positive-de nite variance matrix. 4 It would also be possible to use the full spatial panel GLS tranfromation since the spatial GM procedure also provides estimates of " and. Nevertheless, the tranformed likelihood approach on a model after the spatial Cochrane-Orcutt transformation does not depend on the variance of the inidividual e ects. 7

8 3.. Instrumental Variable Estimator of the Slope Coe cients To be able to de ne the IV estimator, it turns out to be convenient to stack the model di erently. The model is: y it y i;t + u it (9) where y it and u it are m vectors. After taking transpose and staking the observations at di erent times for a given cross-section, we have y i. C A y i. C A mm + u i. C A (2) y it T m y i;t T m u it T m or with the obvious notation Y i Y i; + U i (2) Stacking the cross-sections yields Y Y + U (22) where Y (Y; :::;YN ), Y (Y; ; :::;YN; ) and U (U ; :::;U N ). I de ne the IV estimator of as h i b IV bz b Z bz Y (23) where Z b P H Y with P H H(H H) H where H is vector of instruments used for Y. I suggest the use of the instruments H Y 2 (Y; 2; :::; YN; 2 ) where Y i; 2 (y i; ; :::; y i;t 2 ). However, any instruments that satisfy the following conditions lead to consistent estimates of the spatially correlated disturbances. Assumption 6 The instrument matrix H has a full column rank. Assumption 7 The instruments satisfy the following:. p lim N H H Q HH where Q HH is nite and nonsingular; 2. p lim N H Y Q HY where Q HY is nite and has a full column rank; 3. The instruments H can be expressed as H F(& ; ::; & m ) where each & i is a NT vector of identically and independently distributed random variables and F is an N N nonstochastic absolutely summable matrix. Furthermore, each & i is independent of " it. The rst two assumptions guarantee that the instruments are not degenerate and that they are asymptotically correlated with the variables they replace. The last assumption implies that the instruments are not correlated with the error terms and that a central limit theorem for triangular arrays of quadratic forms can be applied. Given these additional assumptions, the IV estimation produces N 2 consistent estimates. Note that the rate of 8

9 convergence is important for consistency of estimation (the degree of spatial correlation in the residuals) in the the second step of the procedure. Observe that our suggested instruments meet the required conditions. By backward substitution we can eliminate the lagged dependent variables and express the instruments as a function of lagged disturbance terms and lagged explanatory variables. It is easily veri ed that! Xt 2 y t (I mn W) (I N ) j " t j + (I N ) t [" (I m ) ] (24) and hence we have that j H (Y ; 2; :::; Y N; 2) (25) F [" (I ) ; " 2 ; :::; " T 2 ] (26) where F hi T (I mn W ) i IT (I mn W) Our assumptions on the spatial weighting matrices imply that the mnt mnt matrix F is absolutely summable. 3.2 Estimation of the Degree of Spatial Autocorrelation The second step in the proposed estimation procedure is to use moment conditions based on the estimated disturbances: bu t y t I N b IV y t (27) where b IV is the IV estimators of. Kelejian and Prucha (999) show consistency of a similar two stage procedure for univariate single cross-section model with spatial lags in both the dependent variable as well as the error term. Kapoor et al. (27) extend the results for a univariate static panel model. Note that both of these papers consider nonstochastic exogenous variables and hence their results are not directly applicable to the panel VAR model considered here. However, Mutl (26) contains a straightforward extension of their proofs for univariate panel autoregressive models. Hence I conjecture that the spatial GM procedure will also be consistent in a multivariate setting (under an appropriate set of assumptions). To be able to describe the multivariate version of the spatial GM estimation procedure, it proves to be convenient to stack the model di erently. It is also possible to impose more structure on the innovations " it and, in particular, consider that they are generated from a two-way error component model. Recall that the disturbances of the model are generated from u it w ij u jt + it : (28) j 9

10 I now assume that the innovations have a two-way error components structure it (I m ) i + " it ; (29) where the elements of the m vectors i and " it are independent with E [(I m ) i i (I m )] and E (" it " it) ". We can now stack the disturbances and innovations over the di erent cross-sections. In contrast to Section 2, I de ne eu t Nm (u t ; ::; u Nt ) ; e t ( t ; ::; Nt ) ; (3) Nm eu NT m (eu ; ::; eu T ) ; e NT m (e ; ::; e T ) : I additionally de ne the notation for the spatial lag as u it P N j w iju jt and it P N j w ij jt. The stacked spatially lagged disturbances and innovations hence become eu t (u t ; ::; u Nt ) ; e t ( t ; ::; Nt ) ; (3) eu eu ; ::; eu T ; e e ; ::; e T : The multivariate version of the spatial GM estimation is based on the following moment conditions (see the Appendix B for their derivation): E N (T ) e Q e " ; (32) E N (T ) e Q e N w ij " wij; E N (T ) e Q e N i j w ii " ; i E N e Q e ; E N e Q e N N X i w ij wij; j E X N N e Q e N w ii ; where Q and Q are the within and between transformation matrices de ned as and " + T. i Q I T T J T IN ; Q T J T I N ; (33)

11 To express the above moment conditions in terms of the disturbances eu I note that where eu eu ; ::; eu T with eu t conditions then can be written as e eu eu; and e eu eu, (34) u t ; ::; u Nt and u it P N j w iju jt. The six moment ; 2 ; vec ( " ) ; vec ( ) ; (35) where with (i ; ) i i 2 i ; N (T ) i vec Eeu Q i eu + Eeu Q i;n eu ; i 2 N (T ) i vec Eeu Q i eu + Eeu NQ i eu N (T ) i vec Eeu Q i eu + Eeu Q i eu ; i 22 ; i ; (36) 7 5 N (T ) i vec Eeu Q i eu ; N (T ) i vec Eeu Q i eu ; N (T ) i vec Eeu Q i eu ; i 3 I m 2; i i 23 N i 33 N i j N (T ) i vec (Eeu Q i eu) ; (37) w ij w ij ; i 2 (I m w ii ) ; i 3 i N (T ) i vec Eeu Q i eu ; N (T ) i vec Eeu Q i eu : The multivariate spatial GM procedure is based on the sample counterpart of the six moment conditions above. In particular, given an initial estimate, say, b of the slope

12 coe cients, we calculate the projected disturbances (t p; ::; T ) bu it y it yi;t b ; (38) bu it W ij ybu jt ; b u it j W ij ybu jt : j Thus the estimated vectors b eu, eu, b and b eu have dimensions N (T p + ) m where p is the number of lags in the model (in contrast to e.g. eu which has dimensions NT m). However, when the PVAR model only has one lag (p ), as it is for the case considered in this paper, the dimensions do not change. The sample analogue of the moment conditions is then based on G ; 2 ; vec ( " ) ; vec ( ) g &; (39) where the vector & depends on the parameters ; vec ( " ) ; vec ( ) and: with (i ; ) g i g i 2 g i 3 2 G 6 4 N (T ) i vec N (T ) i vec N (T ) i vec i g g 2 g 3 g 2 g 22 g 23 g 3 g 32 g 33 g g 2 g 3 g 2 g 22 g 23 g 3 g 32 g ; g b eu Q i b eu + b eu Qi;N b eu ; g i 2 beu b Q ieu b b + eu NQ ieu ; g22 i b eu b Q ieu b + eu b Qieu ; g32 i g g 2 g 3 g g 2 g 3 3 ; (4) 7 5 beu N (T ) i vec b Qieu ; N (T ) i vec N (T ) i vec beu Q i b eu ; beu Qi b eu ; g3 i I m 2; g i N (T ) i vec beu Qi b eu ; (4) g23 i wij w ij ; g i beu N 2 N (T ) i vec b Qieu ; g i 33 N j (I m w ii ) ; g3 i i N (T ) i vec beu b Qieu : 2

13 The spatial GM procedure can be based on either the rst three sets of (unweighted) moment conditions, or on the full set of (weighted) moment conditions. De ne the vector & and as function of the parameters (analogically to &): & ; vec (" ) G ; 2 ; vec ( " ) g ; (42) where G gij, and g ij;2;3 gi i;2;3. In the rst case, the (unweighted) spatial GM procedure maximizes the objective function & ; vec (" ) & ; vec (" ) ; (43) subject to the fact that the matrix " has to be strictly positive de nite. This is easily implemented by replacing the variance covariance matrices with their Cholesky decompositions and maximizing only with respect to the m (m + ) 2 free parameters in each of the two m m variance covariance matrices. Note that the unweighted spatial GM procedure only uses within-transformed data (using the matrix Q ) and hence does not rely on the random e ects assumption. This then produces suitable initial estimate of for a xed e eects estimation procedure discussed below. If an initial estimate of is desired, it can be obtained by substitution into the fourth moment condition. In the second case, the spatial GM objective function is & ; vec ( " ) ; vec ( ) & ; vec ( " ) ; vec ( ) ; (44) where is a variance covariance matrix of the moment conditions. I follow Kapoor et al. and note that under the normality assumptions becomes ( T " " ) T W ; (45) ( ) T W where T W [T N ; T 2 ; T 3 ], with 2 2NI m 2 T 4 2 P N P N i j w ij w P ij N i [(I m w ii ) + (I m wii)] 3 5 ; (46) T 2 T P N P N i j P N Pi j N i j 2 P N 2 P N P N P N ip j N P N i j w ij w ij w ijw ij w ij w ij ; w ij w ij [(Im w ii ) + (I m wii)] P N i [(I 3 m w ii ) + (I m wii)] w ij w ij [(Im w ii ) + (I m w ii)] (wij w ij ) + wij w ij 7 5 : 3

14 The matrix depends on unknown parameters ( " and ) and hence these have to be replaced by initial consistent estimators in order to make the weighted spatial GM procedure operational. These can be, for example, based on the unweighted spatial GM procedure. 3.3 Quasi Maximum Likelihood (QML) Estimation The likelihood function for the panel VAR model is easily derived under the assumption that " it N(; " ) where " is the m m variance-covariance matrix of " ti. I specify the exact distribution of the initial observations as in Binder et al. (2) and derive the QML function taking this into account. We can de ne the mnt vector Recall that the variance of the initial observations is given by Similarly, we also have that (y ; u 2; ::; u T ) : (47) E (y y ) (I mn W) (I N ) (I mn W ) : (48) E (u t u t) 2 (I mn W) (I N " ) (I mn W ) ; (49) E u t u t (ImN W) (I N " ) (I mn W ) : Thus, we then have that E() and V C() (I mnt W) (I N ) (I mnt W ) ; (5) where W I T W, and " " 2 "... " C A ; (5) " 2 " with being a m m symmetric matrix of parameters (under Assumption NR). The (NR) speci cation leaves the variance-covariance matrix of the initial observations unrestricted - e.g. there are m(m + )2 free additional parameters. The likelihood function for the entire sample is then L N () mnt N log (2) 2 2 ln jj + ln ji mnt Wj (52) 2 tr (I N R ) (I mnt W) I N (I mnt W ) (I N R) S ; where (vech ;vech "; vec ; ) is the vector of parameters. The mt mt matrix R 4

15 is de ned as and the matrix S is R I I m m... I m C A (53) S (y ; :::; y T ) (y ; :::; y T ) : (54) Under the (IOR) or the (UR) conditions, the vector of parameters is composed of only # (vech "; vec ; ) and the likelihood function is as above but with being a function of of " and, as described in Section Computational Issues The computation of the likelihood function should exploit the structure of the [I mt (I N W)] and matrices when evaluating their determinants and inverses. In particular, we can express as (A " ) (A ; (55) " ) (A 2 " ) where A and A 2 are matrices of constants. The inverse of is then D D (A A2 " ) (A2 A " )D D (A2 A " )D (A A2 " ) ; (56) where D A A2 A ". It order to practically implement the likelihood procedure, it is important to specify the analytical gradients of the likelihood function. These are provided in the Appendix. Note that the analytical derivatives speed up the optimization of the maximum likelihood function substantially, especially in the case where the variance covariance matrix of the initial observations is itself function of the remaining parameters of the model (the IOR case). 3.4 Constrained QML Estimation Although the QML estimation based on the likelihood function (52) is feasible, 5 it might become extremely computationally intensive. In this paper, I propose an alternative approach that takes a consistent estimator of the spatial correlation parameter and maximizes a constrained likelihood function. That is, maximize Q N e mnt log (2) 2 h 2 tr R I mnt N 2 ln jj + ln ImNT W b (57) W b I N i I mnt W b RS N ; 5 The QML estimator is likely to be computationally expensive due to the necessity to calculate eigenvalues of a sparse matrix (I mn W) which is of the dimension mn. With large N this becomes a very demanding problem. 5

16 with respect to e (vech ; vech "; vec ), taking the consistent estimator b of as given. The consistent estimator of the spatial correlation be based on the two-step procedure proposed above. Note that the constrained likelihood estimator is equivalent to using the spatial Cochrane-Orcutt transformation (I mn W) and then maximizing the QML function derived under the assumption that the disturbances are independent, i.e. the same as in Binder et al. (25). 4 Monte Carlo Simulations I now turn to the small sample performance of the di erent estimators. To this end I replicate the simulations in Binder et al. (25) who consider the same PVAR model but with independent disturbances. I modify their generation of the disturbances and, in particular, consider the spatial autoregressive speci cation with a spatial weight matrix W and parameter. The speci cation of the spatial weights corresponds to the designs used in Kapoor et al. (27). In particular, I consider three speci cations of W which di er in their degree of sparseness. Each matrix uses a rook design with J 2; 6 or non-zero o -diagonal elements. The parameter takes values in the set f; :25; :5; :9g. I thus have the three different weights matrices and four di erent values of for each of the ve simulation designs considered in Binder et al. (25), i.e. 6 di erent simulation designs in total. For each parameter design, I consider four di erent sample sizes given by combinations of T 2 f3; g and N 2 f5; 25g. In each simulation design and sample size, I use the same VC matrix for the innovations and the individual e ects, i.e. set " and draw the random variables from a normal distribution. 6 As a robustness check, results are available upon request that use di erent ratio of the two variances (denoted by ) as well as alternative distributions (chi-square and student-t). Each simulation design and sample size is replicated, times and the resulting estimates are saved. Tables provides the biases and root-mean-square errors of the di erent estimators. The estimators considered in the experiments are the same estimators as considered in Binder et al. (25), i.e. the four initial GMM estimators as well as the FE-QML derived under the assumption that the disturbances are independent. Additionally I report results for the three-step procedures described in this paper, i.e. the constrained likelihood approach based on spatial GM estimator of the parameter, which is in turn based on an initial GMM estimation. In particular, the estimator labeled GMMs uses the standard moment conditions as suggested by Arellano and Bond (99); estimator labeled GMMe appends these moment conditions by initialization restrictions as proposed by Arellano and Bover (995) and Blundell and Bond (998); while estimator GMMe2 uses the standard Arellano and Bond (99) orthogonality conditions appended by homoscedasticity conditions suggested in Ahn and Schmidt (995, 997); and nally the estimator labeled GMMe3 uses the standard orthogonality conditions appended by both the initialization and homoscedasticity restrictions. 6 Note that it has now been well documented that the performance of the GMM estimators deteriorates with increasing the ratio of variance of the individual e ects to the variance of the innovations, while the QML procedure is invariant to this ratio. The setup in this paper sets this ratio to one and hence sets the odds in favor of the GMM procedures. 6

17 Please see Binder et al. (25) for de nition and more detailed discussion of these estimators in a multivariate context. Next estimator, labeled QMLiid, uses the incorrectly speci ed likelihood function under the assumption of independent disturbances, while the estimator labeled QMLco, uses the same likelihood estimation but with data transformed by the spatial Cochrane-Orcutt transformation based on estimates from spatial GM estimation which is in turn based in estimated disturbances from the GMMe procedure. To save space only results for one particular spatial weights matrix are reported (J 6, meaning that the matrix W has 6 non-zero o -diagonal elements). The designs are as follows: Design : Stationary PVAR with maximum eigenvalue of.6 :4 :2 :2 :4 ; " :7 :5 :5 :7 : Design 2: Stationary PVAR with maximum eigenvalue of.8 :6 :2 :7 :2 ; :2 :6 " :2 :7 : Design 3: Stationary PVAR with maximum eigenvalue of.95 :7 :25 :8 :5 ; :25 :7 " :5 :8 : Design 4: PVAR with unit roots (but not cointegrated) :8 :5 ; " :5 :8 : Design 5: Cointegrated PVAR :5 : :5 : :5 :3 ; " :3 :5 : Table about here. The performance of all estimators deteriorates with increased degree of spatial autocorrelation. This is due to the fact that as increases the expected R 2 of the regression decreases. However, not all estimators are sensitive in the same way. First, note that the QML procedure using the Cochrane-Orcutt transformed data (QMLco) is overall the best performer (in terms of RMSE) when spatial autocorrelation is present. However, the QMLco also shows almost no loss of e ciency relative to the QMLiid procedure even for the cases where there is no spatial autocorrelation ( ). Furthermore, note that the performance of the QMLco estimator is sensitive to the degree of spatial autocorrelation only in the presence of unit roots and/or cointegration. The unit root and cointegrated designs (design 4 and 5) are also the only cases where the extended GMM procedures perform better than the QML procedure, provided that there is no or minimal amount of spatial autocorrelation. 7

18 Next, note that the extended GMM and the QMLiid estimators perform reasonably well even in the presence of spatial autocorrelation. Their performance in terms of RMSE is about the same and deteriorates as increases in absolute value. Additional results with higher show that the QMLiid procedure starts dominating the extended GMM estimators, analogically to the results for in Binder et al. (25). Finally, I note that our simulations document that the performance of the standard GMM procedure deteriorates rapidly with presence of autocorrelation in either dimension (time or space). The GMMs estimator breaks down in presence of unit roots and/or cointegration (as was documented in other studies) but also in presence of high degree of spatial autocorrelation. Therefore, it is interesting to note that the performance of the QMLiid estimator is no worse and often better than that of the extended GMM estimators. This is somewhat surprising because the presence of spatial autocorrelation invalidates the independence assumption on which the QMLiid estimator relies. On the other hand, the moment conditions of the GMM estimators remain valid even if spatial autocorrelation is present. Table 2 gives the size and power properties of hypothesis tests based on the di erent estimation procedures. To save space I only report hypotheses tests concerning the rst element of vec. Results for the remaining elements are similar and are available upon request. The results show that despite satisfactory performance in terms of RMSE, the GMM estimators fail to provide adequate con dence intervals even in the absence of spatial autocorrelation. The nominal size of the hypothesis tests for the extended GMM estimators is between 8 and percent instead of the correct size of 5 percent even in the largest sample (N 25, T ) whereas the nominal size of the QML estimators remains between 5 to 7 percent. An exception is the pure unit root case (design 4) where there is a tendency for overrejection and the nominal size is 8-9 percent for the QML estimators, while the extended GMM estimators have nominal size of 6 to percent. These results replicate the ndings in Binder et al. (25). When spatial autocorrelation is present, all except the QMLco estimator stop providing reasonable con dence intervals and the nominal sizes increase to above 5 percent for :9 even in the largest sample. In contrast, the QMLco estimator remains correctly sized at 5 to 6 percent in this case. With small degree of spatial autocorrelation ( :2), the extended GMM estimators have a nominal size of about 8- percent in the largest sample, while both QML estimators have the sizes 5-7 percent (again with the exception of the pure unit root design). Overall, I conclude that the QMLco estimator provides good small sample guidance regardless of the degree of spatial and temporal correlation in the data. The other estimators break down when the degree of spatial autocorrelation is high but, however, the QMLiid estimator is robust to small amounts of spatial autocorrelation. Table 2 about here. Since this paper extends the spatial GM procedure to the multivariate context, I also report in Table 3 the performance of the spatial GM estimation. The results show that spatial GM procedure works well in all sample sizes and data generation designs under consideration. The highest RMSE is :24 when N 5 and T 5 and it drops to below :2 in the largest sample size (N 25, T ). 8

19 Table 3 about here. 5 Conclusion This paper develops an estimation approach for a panel VAR model with spatial dependence. I extend the literature in several aspects. First, it is studied how cross-sectional dependence of a particular form a ects the various panel VAR estimators. Secondly, I generalize the spatial GM procedure to the multivariate context. This paper proposes a three-step estimation procedure. In the rst step, instrumental variables procedure is used to consistently estimate the spatially correlated disturbances. In the second step, a method of moments estimation is used to obtain a consistent estimate of the spatial parameter. The nal step of the procedure is either a constrained maximum likelihood procedure or moments estimation based on a model transformed by a spatial Cochrane-Orcutt transformation. Finally, the small sample properties of the di erent estimation procedures are studied in a Monte Carlo study. The results show that the constrained likelihood procedure works well in small samples. They also document that the QML estimation based on the independence assumption is robust to small amount of spatial autocorrelation in the data. In future research, it would also be of interest to prove asymptotic normality of the proposed estimator as well as to derive the asymptotic properties of the QML estimator under some reasonable set of assumptions. An interesting complement to the approach of this paper would be to develop estimation procedures for panel VAR models with alternative speci cations of cross-sectional correlation (e.g. nonparametric, or factor models) and compare the their relative performance under di erent data generating designs. 9

20 A Appendix - Derivatives of the QML Function To speed up computation, I derive analytical expressions for the partial derivatives of the likelihood function L N (). The rst di erential is dl N N 2 tr d tr (I mnt W) dw (A.) + 2 tr (I N R ) (I mnt W) I N d (I mnt W ) (I N R) S 2 tr (I N dr ) (I mnt W) I N (I mnt W ) (I N R) S 2 tr (I N R ) (I mnt W) I N (I mnt W ) (I N dr) S 2 tr (I N R ) ( dw) I N (I mnt W ) (I N R) S 2 tr (I N R ) (I mnt W) I N ( dw ) (I N R) S N 2 vec D mt dvech vec (I mnt W ) vecw d + 2 vec I N (I mnt W ) (I N R) S (I N R ) (I mnt W) I N vec (I N d) 2 vec S (I N R ) (I mnt W) I N (I mnt W ) vec (IN dr ) 2 vec (I mnt W) I N (I mnt W ) (I N R) S vec (IN dr) + 2 tr (I N R ) W I N (I mnt W ) (I N R) S d + 2 tr (I N R ) (I mnt W) I N W (I N R) S d; where D mt is a duplication matrix (such as that D k vech(x) vec(x) for any k k matrix X), K sq is a commutation matrix (such that K sq vec(x) vec(x ) for any s q matrix X). Hence dl N 2 vec I N HD mt dvech tr (I mnt W) W d (A.2) + 2 vec I N (I mnt W ) (I N R) S (I N R ) (I mnt W) I N HD mt vech 2 vec S (I N R ) (I mnt W) I N (I mnt W ) HKmT;mT dvecr 2 vec (I mnt W) I N (I mnt W ) (I N R) S H dvecr +tr (I N R ) W I N (I mnt W ) (I N R) S d; 2

21 where the matrix of constants H is given by H [(I N K mt;n ) (veci N ) I mt ] I mt : (A.3) The di erential of the inverse of the variance covariance matrix under condition (IOR) is dvech vech [(A d ) + (A 2 d)] (A.4) D mt (I T K m;t I m )(veca I m 2)D m dvech + +D mt (I T K m;t I m )(veca 2 I m 2)D m dvech D mt B D m dvech + D mt B 2D m dvech and the di erential of the matrix R containing the slope coe cients is dvecr vec(a 3 d) (A.5) (I T K m;t I m )(veca 3 I m 2)dvec B 3 dvec with A, A 2, A 3, B, B 2 and B 3 being matrices of constants re ecting the structure of and R. In particular,. A C (A.6)... A and De ning M I N A 3 A 2 I T A (A.7) ::: + I N (I mnt W ) (I N R) S (I N R ) (I mnt W) I N ; M 2 (I mnt W) I N (I mnt W ) (I N R) S; M 3 (I N R ) W I N (I mnt W ) (I N R) (I mnt W) W;. C A (A.8) (A.9) 2

22 we can write the Jacobian of L N () in a partitioned form as 2 [vec (M )] 3 HB D m : DL N () 6 [vec (M )] HB 2 D m : [vec (M 2 )] H (I + K mt;mt ) B 3 : 5 ; (A.) 2tr (M 3 ) where : denotes horizontal stacking. Under the (IOR) condition the di erential of the variance covariance matrix of the initial observations becomes dvech dvech " (A.) +Dm [(I m ) (I m )] [I m 2 ( )] D m dvech " Dm [d (I m )] [I m 2 ( )] vec " Dm [(I m ) d] [I m 2 ( )] vec " Dm [(I m ) (I m )] [I m 2 ( )] [(d ) + ( d)] [I m 2 ( )] vec " dvech " + Dm [(I m ) (I m )] [I m 2 ( )] D m dvech h i " (vec " ) [I m 2 ( )] Dm vec [d (I m )] h i (vec " ) [I m 2 ( )] Dm vec [(I m ) d] h (vec " ) [I m 2 ( )] Dm [(I m ) (I m )] [I m 2 ( )] vec [(d ) + ( d)] i dvech " + Dm [(I m ) (I m )] [I m 2 ( )] D m dvech h i " (vec " ) [I m 2 ( )] Dm (I m K m I m ) [I m 2 vec (I m )] dvec h i (vec " ) [I m 2 ( )] Dm (I m K m I m ) [vec (I m ) I m 2] dvec h i (vec " ) [I m 2 ( )] Dm [(I m ) (I m )] [I m 2 ( )] (I m K m I m ) [(I m 2 ) + ( I m 2)] dvec Jacobian of then becomes 2 DL N (#) [vec (M )] 3 H (B C + B 2 ) D m : 4 [vec (M )] HB C 2 + [vec (M 2 )] H (I + K mt;mt ) B 3 : 5 ; (A.2) 2 2tr (M 3 ) 22

23 where C I m 2 + [(I m ) (I m )] [I m 2 ( )] (A.3) h i C 2 (vec " ) [I m 2 ( )] Dm (I m K m I m ) [I m 2 vec (I m )] h i (vec " ) [I m 2 ( )] Dm (I m K m I m ) [vec (I m ) I m 2] h i (vec " ) [I m 2 ( )] Dm [(I m ) (I m )] [I m 2 ( )] (I m K m I m ) [(I m 2 ) + ( I m 2)] B Appendix - Derivation of the Moment Conditions Observe rst that Q e I T J T T IN [(T e) + e"] (B.) Q e" + I T J T T IN (T e) Q e" + I T T J T T I N e Q e" + I T T T J T T e Q e" + T T T T e Q e": The rst moment condition is based on the following observation: Ee Q e Ee Q Q e Ee" Q e" Ee" I T J T T IN e" (B.2) Ee" e" E T e" ; ::; e" T (JT I N ) e" ; ::; e" T! Ee" t e" t e" t; ::; e" t e" ; ::; e" T T t Ee" it e" it i E T T NT " T E t t t t Ee" t e" s s e" t e" s s NT T N " N (T ) " T 23

24 The second moment conditions follows from Ee Q e Ee" Q e" Ee" IT J T T IN e" (B.3) (J T I N ) Ee" e" e" E T ; ::; e" T t t Ee" t e" t T E" it " it i (T ) (T ) (T ) t T s t Ee" t e" s s e" ; ::; e" T E" it " is (T ) i E" it " it (w i ; ::; w in ) E (" t; ::; " Nt) (" t; ::; " Nt) (w i ; ::; w in ) i (w i ; ::; w in ) (I N " ) (w i ; ::; w in ) i i w ij " wij: j The third moment condition is based on Ee Q e Ee" Q e" Ee" IT J T T IN e" (B.4) (J T I N ) e" ; ::; e" T Ee" e" e" E T ; ::; e" T t t Ee" t e" t T E" it " it i (T ) (T ) t T s t Ee" t e" s s i E" it " is (T ) i (w i ; ::; w in ) E (" t; ::; " Nt) " it i (w i ; ::; w in ) (e i;n " ) (T ) i E" it " it i w ii " ; where we use e i;n to denote an N vector of zeros with an entry of one on the i th position. To derive the next set of moment conditions involving Q, we note that Q e T J T I N [(T e) + e"] (B.5) Q e" + T J T I N (T e) Q e" + T J T T e Q e" + ( T I N ) e: i 24

25 Furthermore, denoting fw [(w ; ::; w N ) ; ::; (w N ; ::; w NN )] ; (B.6) we have that 2 (I Q e Q e" + J N ) ; ::; (I N N 6 ) 3 7 T T I N 4. 5 W f (B.7) (I N ) ; ::; (I N N ) Q e" + J T T I N T [(I N ) ; ::; (I N N)] W f Q e" + J T T T [(I N ) ; ::; (I N N)] W f Q e" + T [(I N ) ; ::; (I N N)] W f Q e" + T e : Thus we have Ee Q e E [e" + ( T I N ) e] J T T I N [e" + (T I N ) e] Ee" J T T I N e" + E e ( T I N ) J T T I N (T I N ) e E T e" ; ::; e" T (JT I N ) e" ; ::; e" T + E e T T J T T I N e! E e" T t; ::; e" ; ::; e" T + T E e e t t Ee" t T e" s + T E e e T t s T N " + T N N : T e" t t s Ee" it e" is + T i E e i e i i (B.8) 25

26 Furthermore, Ee Q e E h e" + ( T I N ) ei J h i T T I N e" + ( T I N ) e Ee" J T T I N e" + E e ( T I N ) J T T I N (T I N ) e e" E T ; ::; e" T (J T I N ) e" ; ::; e" T + E e T T J T T I N e! E e" T t; ::; e" t e" ; ::; e" T + T E e e T T +T t t t t Ee" t e" s + T E e e T s t s Ee" it e" is + T i E e i e i (w i ; ::; w in ) E (" t; ::; " Nt) (" t; ::; " Nt) (w i ; ::; w in ) i (w i ; ::; w in ) E ( ; ::; N) ( ; ::; N) (w i ; ::; w in ) i (w i ; ::; w in ) (I N " ) (w i ; ::; w in ) i +T i (w i ; ::; w in ) (I N ) (w i ; ::; w in ) i w ij " wij + T j i w ij wij j i i w ij wij; j (B.9) 26

27 and Ee Q e E h e" + ( T I N ) ei T J T I N [e" + (T I N ) e] Ee" J T T I N e" + E e ( T I N ) J T T I N (T I N ) e e" E T ; ::; e" T (J T I N ) e" ; ::; e" T + E e T T J T T I N e! E e" T t; ::; e" ; ::; e" T + T E e e T T + t t i t t e" t Ee" t e" s + T E e e T s i E (" t; ::; " Nt) " it i E ( ; ::; N) e i i w ii " + T i w ii i t w ii : i s Ee" it e" is + T i E e i e i i (B.) 27

28 References [] Ahn, S.C. and P. Schmidt, 995, E cient Estimation of Models for Dynamic Panel Data, Journal of Econometrics 68, [2] Anselin, L., 988, Spatial Econometrics: Methods and Models, Kluwer Academic Publishers, Boston. [3] Arellano, M. and S.R. Bond, 99, Some Tests of Speci cation for Panel Data: Monte Carlo Evidence and Application to Employment Equations, Review of Economic Studies 58, [4] Arellano, M. and O. Bover, 995, Another Look at the Instrumental Variable Estimation if Error-Component Models, Journal of Econometrics 68, [5] Binder, M., Ch. Hsiao and M. H. Pesaran, 25, Estimation and Inference in Short Panel Vector Autoregressions with Unit Roots and Cointegration, Econometric Theory 2, [6] Chen, X. and T. Conley, 2, A New Semiparametric Spatial Model for Panel Time Series, Journal of Econometrics 5, [7] Cli, A. and J. Ord, 973, Spatial Autocorrelation, Pion, London. [8] Cli, A. and J. Ord, 98, Spatial Processes, Models and Applications, Pion, London. [9] Conley, T., 99, GMM Estimation with Cross Sectional Dependence, Journal of Econometrics 92, -45. [] Hsiao, C., M. H. Pesaran and A. K. Tahmiscioglu, 22, Maximum Likelihood Estimation of Fixed E ects Dynamic Panel Data Models Covering Short Time Periods, Journal of Econometrics 9, 7-5. [] Kapoor M., H. Kelejian and I.R. Prucha, 27, Panel Data Models with Spatially Correlated Error Components, Journal of Econometrics 4, [2] Kelejian, H.H. and I.R. Prucha, 998, A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances, Journal of Real Estate Finance and Economics 7, [3] Kelejian, H.H. and I.R. Prucha, 999, A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model, International Economic Review 4, [4] Kelejian, H.H. and I.R. Prucha, 24, Estimation of Simultaneous Systems of Spatially Interrelated Cross Sectional Equations, Journal of Econometrics 8, [5] Kelejian, H.H. and I.R. Prucha, 2, On the Asymptotic Distribution of the Moran I Test Statistic with Applications, Journal of Econometrics 4, [6] Mulaik, S.A., 972, The Foundations of Factor Analysis, McGraw-Hill, New York. 28

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30 Table : Bias and RMSE of Phi_ Bias RMSE J2 Design N lambda,2,5,9,2,5,9,2,5,9,2,5,9 T5; GMMs,2,2,29,59,3,4,34,289,84,86,3,276,2,23,243,475 GMMe -,8 -,8 -, -,53 -,37 -,39 -,54 -,47,63,64,76,84,48,49,67,3 GMMe2,7,7,,56,4,4,45 -,6,72,73,87,96,66,67,83,3 GMMe3 -,8 -,8 -, -,65 -,45 -,48 -,65 -,56,63,64,75,88,5,52,7,33 QMLiid,,,,5,5,4,3,26,56,57,67,64,7,8,38,335 QMLco,,,,,4,4,4,2,56,56,56,56,8,8,8,9 T; GMMs,8,9,26,,96,97,7,65,47,48,58,62,39,4,54,27 GMMe -, -,2 -,7 -,7 -,69 -,7 -,89 -,8,37,38,47,5,2,4,32,255 GMMe2,7,8,,37,54,54,5 -,85,4,4,48,6,7,8,4,26 GMMe3 -, -, -,7 -,2 -,74 -,76 -,96 -,83,37,38,47,55,9,2,38,257 QMLiid -, -, -,2 -,3,4,4,4,2,3,3,38,97,72,73,87,26 QMLco -, -, -, -,,4,4,3,3,3,3,3,3,72,72,72,72 J2 Design 2 N lambda,2,5,9,2,5,9,2,5,9,2,5,9 T5; GMMs,33,34,48,245,77,83,223,42,84,85,2,33,256,262,34,528 GMMe -,5 -,5 -,5 -,22 -,2 -,2 -,6 -,5,49,49,57,24,6,6,9,89 GMMe2,4,4,9,89,68,7,82,68,64,65,76,83,5,53,73,24 GMMe3 -,5 -,5 -,6 -,32 -,5 -,6 -,22 -,58,49,49,57,26,7,7,2,9 QMLiid,,,2,7,,,4,7,43,43,5,26,98,,7,26 QMLco,,,,,8,8,8,8,43,43,43,43,98,98,98,99 T; GMMs,3,3,4,48,3,32,45,2,46,46,57,75,58,59,7,26 GMMe -,6 -,7 -, -,66 -,42 -,43 -,54 -,97,27,27,33,93,76,76,86,5 GMMe2,3,3,7,6,7,7,72 -,5,33,33,39,99,,,5,38 GMMe3 -,6 -,7 -, -,68 -,44 -,45 -,57 -,98,27,27,33,96,8,8,9,52 QMLiid,,,,2,5,5,5,3,2,22,26,66,5,5,58,47 QMLco,,,,,5,5,5,4,2,2,2,2,5,5,5,53

31 Table (cont.): Bias and RMSE of Phi_ Bias RMSE J2 Design 3 N lambda,2,5,9,2,5,9,2,5,9,2,5,9 T5; GMMs,9,92,23,363,32,324,358,475,84,87,239,54,496,498,537,644 GMMe,3,3,2 -, -,7 -,7 -, -,37,59,59,68,26,5,6,25,68 GMMe2,9,9,23,86,76,77,84,68,8,8,95,23,82,8,89,227 GMMe3,2,2, -,8 -, -,2 -,7 -,4,58,58,66,22,2,3,2,65 QMLiid,4,4,3,7,2,2,6,72,5,5,6,43,9,,28,294 QMLco,4,4,4,4,,,,6,5,5,5,5,,,,7 T; GMMs,58,6,79,25,99,2,23,263,87,9,2,26,245,247,26,33 GMMe -,3 -,3 -,5 -,43 -,28 -,29 -,37 -,65,33,34,39,76,66,66,73,2 GMMe2,6,7,22,7,77,78,79,8,44,44,53,7,5,6,2,32 GMMe3 -,3 -,3 -,5 -,46 -,3 -,32 -,4 -,66,33,34,39,78,68,69,75,23 QMLiid,2,2,2,8,,,3,53,25,25,3,75,57,58,68,66 QMLco,2,2,2,2,,,,,25,25,25,25,57,57,57,57 J2 Design 4 N lambda,2,5,9,2,5,9,2,5,9,2,5,9 T5; GMMs,79,793,82,777,8,86,778,694,95,96,97,884,97,96,894,89 GMMe,,,2,4,7,7,9,24,25,26,3,7,56,55,62,9 GMMe2,8,8,3,72,5,52,67,86,45,46,59,66,3,3,53,86 GMMe3,,,,9,8,7,,24,25,25,3,6,52,53,6,87 QMLiid,6,5,8,32,2,9,25,92,48,5,5,9,9,94,3,242 QMLco,4,5,3,4,9,9,22,8,72,58,3,64,3,93,92, T; GMMs,528,529,52,437,498,495,476,396,569,569,56,47,535,532,54,452 GMMe,,,,3,4,4,4,,2,2,4,26,24,24,25,44 GMMe2,3,3,4,44,4,4,46,4,9,2,24,78,73,74,78,83 GMMe3,,,,4,5,5,5,,2,2,3,25,24,24,25,44 QMLiid,2,2,2,3,2,2,5,58,2,25,3,58,48,5,56,38 QMLco,3,2,3 -,2,2,,2,9,2,26,3,72,48,62,52,66