P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 A Unified Estimation Approach for Spatial Dynamic Panel Data Models 415 TABLE 14.3 Performance of Estimators When the DGP Is Explosive TnEstimator ␥␳␤␭␴ 2 No Time Dummy in the DGP (Equation 14.26): (1) 10 54 A Bias 0.0053 0.0395 0.0049 −0.0336 −0.0241 SD 0.0336 0.0584 0.0465 0.0422 0.0626 RMSE 0.0340 0.0705 0.0467 0.0540 0.0670 CP 0.9200 0.8890 0.9270 0.8630 0.9230 10 54 Unified Bias −0.0018 0.0031 −0.0007 −0.0196 −0.0360 SD 0.0379 0.1382 0.0504 0.1201 0.0716 RMSE 0.0380 0.1382 0.0504 0.1217 0.0801 CP 0.9170 0.9310 0.9270 0.9100 0.8070 (2) 50 18 A Bias ****** ****** 2.4973 −0.0624 ****** SD ****** ****** 264.78 0.2958 ****** RMSE ****** ****** 264.79 0.3023 ****** CP 0.0150 0.0090 0.0140 0.0130 0.0110 50 18 Unified Bias −0.0013 −0.0013 −0.0025 −0.0088 −0.0065 SD 0.0246 0.0931 0.0373 0.0878 0.0543 RMSE 0.0246 0.0931 0.0374 0.0882 0.0547 CP 0.9480 0.9440 0.9420 0.9260 0.9050 (3) 50 54 A Bias ****** ****** −4.1263 −0.0668 ****** SD ****** ****** 724.64 0.3096 ****** RMSE ****** ****** 724.66 0.3167 ****** CP 0.0010 0.0000 0.0000 0.0010 0.0000 50 54 Unified Bias −0.0004 −0.0006 0.0002 −0.0005 −0.0016 SD 0.0139 0.0557 0.0203 0.0510 0.0315 RMSE 0.0139 0.0557 0.0203 0.0510 0.0315 CP 0.9450 0.9380 0.9600 0.9250 0.9130 Time dummy in the DGP (Equation 14.27): (1) 10 54 B Bias 0.0021 0.0386 0.0037 −0.0305 −0.0257 SD 0.0346 0.0635 0.0482 0.0462 0.0639 RMSE 0.0347 0.0743 0.0483 0.0554 0.0689 CP 0.9190 0.8870 0.9240 0.8880 0.9100 10 54 Unified Bias −0.0049 0.0029 −0.0003 −0.0191 −0.0371 SD 0.0390 0.1435 0.0529 0.1200 0.0688 RMSE 0.0394 0.1435 0.0529 0.1216 0.0782 CP 0.9120 0.9060 0.9230 0.9090 0.8090 (2) 50 18 B Bias ****** ****** −4.0205 −0.0478 ****** SD ****** ****** 105.34 0.2891 ****** RMSE ****** ****** 105.41 0.2931 ****** CP 0.1030 0.0640 0.0960 0.0790 0.0660 50 18 Unified Bias −0.0011 0.0014 −0.0030 −0.0033 −0.0061 SD 0.0248 0.0972 0.0378 0.0885 0.0536 RMSE 0.0248 0.0972 0.0379 0.0885 0.0540 CP 0.9520 0.9390 0.9430 0.9260 0.9110 (3) 50 54 B Bias ****** ****** −35.49 −0.0596 ****** SD ****** ****** 835.56 0.3128 ****** RMSE ****** ****** 836.31 0.3184 ****** CP 0.0020 0.0000 0.0010 0.0040 0.0000 50 54 Unified Bias −0.0001 −0.0009 −0.0001 −0.0030 −0.0031 SD 0.0143 0.0553 0.0215 0.0521 0.0308 RMSE 0.0143 0.0553 0.0215 0.0522 0.0310 CP 0.9410 0.9370 0.9380 0.9220 0.9270 Note: 1. ␪ 0 = (0.4, 0.4, 1, 0.4, 1)  . 2. ****** denotes an explosive number, which is of the order 10 11 for the column of ␴ 2 , and 10 5 for other columns. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 416 Handbook of Empirical Economics and Finance 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=1% 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=5% 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=1% 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=5% Note: 1. denotes the power curve for T = 10, and —— denotes the power curve for T = 50. 2. The first row is for the two-sided tests and the second row is for the one-sided tests. FIGURE 14.1 Power curves under the unified approach for H 0 : ␥ 0 + ␳ 0 + ␭ 0 = 1. unified approach to get the power curves. The results are in Figure 14.1. For the two-sided tests, the sum ␭ 0 + ␥ 0 + ␳ 0 under the alternative hypothesis ranges from 0.65 to 1.35 with a 0.7 200 increment; for the one-sided test with H 1 : ␭ 0 + ␥ 0 + ␳ 0 < 1, the sum ␭ 0 + ␥ 0 + ␳ 0 ranges from 0.65 to 1.0 with a 0.35 200 increment. From Figure 14.2, we can see that the empirical sizes 18 are close to the theoretical ones and the tests are more powerful when T = 50 than those for the small T = 10. The power seemsreasonable for thelarge T = 50.We run additional simulations where we use the corresponding estimation method without any transformation. Figure 14.2 is the counterparts 19 of Table 14.1. 18 For the empirical size, the T = 10 case has 2.4%, 2.2%, 9.1%, and 8.8% from the first row to the second row, and the T = 50 case has 1.6%, 1.7%, 6.5%, and 5.8%. As the significance level are 1%, 1%, 5%, and 5% correspondingly, a larger T will yield empirical sizes closer to the theoretical values. 19 For the first row in Table 14.2, when the sum ␭ 0 + ␥ 0 + ␳ 0 is much larger than 1 (i.e., the process is explosive), the estimates might not be available due to overflow without the unified transformation. Hence, for the two-sided power curves, we allow the sum only up to 1.3. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 A Unified Estimation Approach for Spatial Dynamic Panel Data Models 417 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α =1% 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α =5% 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=1% 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sum α=5% Note: 1. denotes the power curve for T = 10, and —— denotes the power curve for T = 50. 2. The first row is for the two-sided tests and the second row is for the one-sided tests. FIGURE 14.2 Power curves under Yu, de Jong, and Lee (2007) for H 0 : ␥ 0 + ␳ 0 + ␭ 0 = 1. We can see that, when ␭ 0 +␥ 0 +␳ 0 < 1, the test is more powerful by using the corresponding method without any transformation; when ␭ 0 + ␥ 0 + ␳ 0 > 1, the power curves are irregular and we need to rely on the unified approach for the inferences. 20 14.5 Conclusion This chapter establishes asymptotic properties of QMLEs for SDPD models with both time and individual fixed effects when both the number of individ- uals nandthenumberoftimeperiods T can be large.Insteadofusingdifferent 20 For the empirical size, the T = 10 case has 34.8%, 0.3%, 44.9%, and 1.5% from the first row to the second row in Table 14.2, and the T = 50 case has 1.1%, 0.8%, 4%, and 4%. Hence, when T is small, the empirical sizes could be far away from the theoretical values. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 418 Handbook of Empirical Economics and Finance estimation methods depending on whether the DGP has time effects or not and whether the DGP is stable or not, we propose a data transformation ap- proachtoeliminateboththetimeeffectsandthepossibleunstableorexplosive effects. The transformation is motivated by the possible co-integration rela- tionship in the SDPD model, which is implied by the unit eigenvalues in the spatial weights matrix W n . Unlike the co-integration in the multi-variate time series, the co-integrating vector is known and does not need to be estimated. With the proposed data transformation, the possible unstable or explosive components and time effects can be eliminated. Thetransformationusestheco-integratingmatrix.Theeffectivesamplesize n ∗ after transformation corresponds to the co-integration rank, which is the number of eigenvalues not equal to the unity. This transformation isof partic- ular value when the process may contain explosive roots, as usual estimation methods can be poorly performed under such a situation. For the unified ap- proach, when T is relatively larger than n ∗ , the estimators are √ n ∗ T consistent and asymptotically centered normal; when n ∗ is asymptotically proportional to T, the estimators are √ n ∗ T consistent and asymptotically normal, but the limit distribution is not centered around 0; when T is relatively smaller than n ∗ , the estimators are consistent with rate T and have a degenerate limit dis- tribution. We also propose a bias correction for our estimators. We show that when T grows faster than n ∗1/3 , the correction will asymptotically eliminate the bias and yield a centered confidence interval. Monte Carlo experiments have demonstrated a desirable finite sample performance of the estimator. A test statistic for testing possible spatial co-integration is also considered. In Lee and Yu (2010b), this unified estimation approach is applied to study the market integration in Keller and Shiue (2007) with the SDPD model and test for the spatial co-integration. Appendices A Some Notes A.1 The Eigenvalues of A n :Three Cases of the DGP From Subsection 14.2.1, the eigenvalues matrix of A n can be decomposed as D n = ␥ 0 +␳ 0 1−␭ 0 J n + ˜ D n , where J n = diag{1 m n , 0, ···, 0} and ˜ D n = diag{0, ···, 0, d n,m n +1 , ···,d nn } with |d ni | < 1. Hence, A h n = ( ␥ 0 +␳ 0 1−␭ 0 ) h R n J n R −1 n + B h n with B h n = R n ˜ D h n R −1 n .Asd ni = ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni , the derivative of d ni = ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni as a function of ␻ ni is ∂( ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni ) ∂␻ ni = ␳ 0 +␥ 0 ␭ 0 (1−␭ 0 ␻ ni ) 2 . Thus, d ni is a monotonicfunction of ␻ ni . Our settingas- sumes that |d ni | < 1 whenever d ni = 1. This requirement can be satisfied with appropriaterestriction on the parameter space of ␳ 0 , ␥ 0 and ␭ 0 as shownbelow. The case with ␳ 0 + ␥ 0 ␭ 0 = 0 implies that d ni is a constant function of ␻ ni . As |␭ 0 | < 1 (implied by Assumptions 1 and 3), the derivative is zero if and P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 A Unified Estimation Approach for Spatial Dynamic Panel Data Models 419 only if ␳ 0 + ␥ 0 ␭ 0 = 0, i.e., ␳ 0 =−␭ 0 ␥ 0 .Inthis situation, d ni = ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni = ␥ 0 , and all |d ni | < 1if|␥ 0 | < 1. 21 The d ni is a strictly increasing function of ␻ ni if and only if ␳ 0 + ␭ 0 ␥ 0 > 0; otherwise it is a strictly decreasing function of ␻ ni when ␳ 0 + ␭ 0 ␥ 0 < 0. Let ␥ 0 + ␳ 0 + ␭ 0 = 1 +a, where a is a constant. We have the stable case when ␥ 0 + ␳ 0 + ␭ 0 < 1; the spatial cointegration case when ␥ 0 +␳ 0 +␭ 0 = 1 but ␥ 0 = 1; and the explosive case when ␥ 0 +␳ 0 +␭ 0 > 1. The condition ␳ 0 + ␥ 0 ␭ 0 > 0(< 0) is equivalent to (1 − ␥ 0 )(1 − ␭ 0 ) > −a (< −a) because (1 −␥ 0 )(1 −␭ 0 ) = ␳ 0 + ␥ 0 ␭ 0 − a. Assume that d ni is an increasing function of ␻ ni .AsW n is row-normalized, −1 ≤ ␻ ni ≤ 1 for all i.With the relation d ni = ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni on [−1, 1], d ni = ␥ 0 −␳ 0 1+␭ 0 at ␻ ni =−1, and d ni = ␥ 0 +␳ 0 1−␭ 0 at ␻ ni = 1. Hence, the smallest eigenvalue of A n will be greater than or equal to ␥ 0 −␳ 0 1+␭ 0 , and the largest eigenvalue will occur at ␻ ni = 1. Hence, the possible range of d ni with ␻ ni in [−1, 1] is [ ␥ 0 −␳ 0 1+␭ 0 , ␥ 0 +␳ 0 1−␭ 0 ]. The smallest eigenvalue of A n will be greater than −1if ␥ 0 − ␳ 0 1 + ␭ 0 > −1 ⇔ 1 +␥ 0 + ␭ 0 > ␳ 0 ⇔ 1 − ␳ 0 > − a 2 . Also, whenever ␻ ni < 1−␥ 0 ␳ 0 +␭ 0 , the corresponding d ni < 1. This is so, because the critical value ␻ ∗ such that ␥ 0 +␳ 0 ␻ ∗ 1−␭ 0 ␻ ∗ = 1isat␻ ∗ = 1−␥ 0 ␳ 0 +␭ 0 = 1 − a (␳ 0 +␭ 0 ) . In summary, for any eigenvalue ␻ ni of W n (with |␻ ni |≤1), the correspond- ing eigenvalue of A n is d ni = ␥ 0 +␳ 0 ␻ ni 1−␭ 0 ␻ ni . Under the situation(1−␥ 0 )(1−␭ 0 ) > −a, we have d ni < 1if␻ ni < 1 − a ␳ 0 +␭ 0 ; and d ni > −1if1− ␳ 0 > − a 2 . Hence, we have the following sufficient conditions for three cases in our studies. Assume that |␭ 0 | < 1 and (1 −␥ 0 )(1 −␭ 0 ) > −a. 1. Stable case: a < 0. If ␳ 0 + ␭ 0 > 0, all d ni ≤ 1 (because ␻ ni < 1 − a ␳ 0 +␭ 0 ); if 1 − ␳ 0 > − a 2 , −1 < d ni . 2. Spatial co-integration case: a = 0. When ␻ ni = 1, d ni = 1; when ␻ ni < 1 and 1 − ␳ 0 > 0, then |d ni | < 1. 3. Explosive case: a > 0. When ␻ ni = 1, d ni > 1; when ␻ ni < 1 − a ␳ 0 +␭ 0 = 1−␥ 0 ␳ 0 +␭ 0 , |d ni | < 1; furthermore, with 1 −␳ 0 > − a 2 , |d ni | < 1. A.2 Decomposition From Equation 14.2, by iterative substitution, we have Y nt = A t+1 n Y n,−1 + t  h=0 A h n S −1 n (c n0 + X n,t−h ␤ 0 + V n,t−h + ␣ t−h,0 l n ). 21 For this special case, the model becomes Y nt = ␥ 0 Y n,t−1 +S −1 n (X nt ␤ 0 +c n0 +␣ t0 l n +V nt ). Hence, this case is Y nt = ␥ 0 Y n,t−1 + S −1 n X nt ␤ 0 + ␣ t0 1−␭ 0 l n + ⑀ nt , where ⑀ nt = ␭ 0 W n ⑀ nt + c n0 + V nt has the panel disturbance structure in Kapoor, Kelejian, and Prucha (2007). This model is close to the one considered in Su and Yang (2007) except for the resulting regressor term. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 420 Handbook of Empirical Economics and Finance As S −1 n l n = 1 1−␭ 0 l n and A n = S −1 n (␥ 0 I n + ␳ 0 W n ) = (␥ 0 I n + ␳ 0 W n )S −1 n , using W n l n = l n ,wehave A h n S −1 n l n = 1 1−␭ 0 ( ␥ 0 +␳ 0 1−␭ 0 ) h l n .ByA h n = ( ␥ 0 +␳ 0 1−␭ 0 ) h R n J n R −1 n + B h n and R n J n R −1 n S −1 n = S −1 n R n J n R −1 n = 1 1−␭ 0 R n J n R −1 n (see Proposition B.4 in Yu, de Jong, and Lee 2007), the above equation can be written as Y nt = A t+1 n Y n,−1 + t  h=0 B h n S −1 n (c n0 + X n,t−h ␤ 0 + V n,t−h ) + 1 1 − ␭ 0 t  h=0  ␥ 0 + ␳ 0 1 − ␭ 0  h ×␣ t−h,0 l n + 1 1 − ␭ 0 t  h=0  ␥ 0 + ␳ 0 1 − ␭ 0  h R n J n R −1 n (c n0 + X n,t−h ␤ 0 + V n,t−h ). For A t+1 n Y n,−1 ,wehave A t+1 n Y n,−1 = ( ␥ 0 +␳ 0 1−␭ 0 ) t+1 R n J n R −1 n Y n,−1 +B t+1 n Y n,−1 , where B t+1 n Y n,−1 = ∞  h=t+1 B h n S −1 n (c n0 + X n,t−h ␤ 0 + V n,t−h ) + 1 1 − ␭ 0 ∞  h=t+1 ␣ t−h,0 B h n l n , using B n A n = B 2 n and B n S −1 n = S −1 n B n . The item with B h n l n is zero. Because R n is the eigenvectors matrix of W n and its first column is l n ,wehave R −1 n l n = e n1 which is the first unit vector. As ˜ D n e n1 = 0, it follows that B n l n = 0. Hence, we can decompose Y nt as Y nt = Y u nt + Y s nt + Y ␣ nt , which is Equation 14.3. The Y s nt represents a stable component as the eigenvalues of B n can be less than unity in absolute value for many parameter values (see Appendix A.1). The Y ␣ nt captures the component due to time dummies. As | ␥ 0 +␳ 0 1−␭ 0 | < 1ifand only if −1 < ␥ 0 + ␳ 0 + ␭ 0 < 1 because ␭ 0 < 1, Y u nt is also stable when ␥ 0 +␳ 0 +␭ 0 < 1. But when ␥ 0 +␳ 0 +␭ 0 = 1(> 1), then ␥ 0 +␳ 0 1−␭ 0 = 1 (> 1) and Y u nt may represent the unstable or explosive components. A.3 Data Transformation We can transform Equation 14.1 by I n − W n into Equation 14.4, where the remaining (I n − W n )c n0 can be regarded as the individual effects. A spe- cial feature of the transformed Equation 14.4 is that the variance matrix of (I n − W n )V nt is equal to ␴ 2 0  n ≡ ␴ 2 0 (I n − W n )(I n − W n )  , which is singular. Hence, there is a linear dependence among the elements of (I n − W n )V nt .An effective estimation method shall eliminate the linear dependence. This can be donewith the eigenvalues and eigenvectors decomposition (see,e.g., Theil 1971, Chapter 6). Let [F n ,H n ]bethe orthonormalmatrix of eigenvectors and  n be thediago- nalmatrixofnonzero eigenvalues of  n suchthat n F n = F n  n and n H n = 0. That is, the columns of F n consist of eigenvectors of nonzero eigenvalues and those of H n are for zero-eigenvalues of  n . Let n ∗ be the number of nonzero eigenvalues. The F n is an n ×n ∗ matrix and  n is an n ∗ ×n ∗ diagonal matrix. Thus,  n F n = F n  n ,F  n F n = I n ∗ ,  n H n = 0,H  n H n = I n−n ∗ , F  n H n = 0,F n F  n + H n H  n = I n ,F n  n F  n =  n . (14.28) P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 A Unified Estimation Approach for Spatial Dynamic Panel Data Models 421 Because  n H n = 0, it implies that (I n − W n )  H n = 0. In turn, W n (I n − W n ) = W n (F n F  n +H n H  n )(I n −W n ) = W n F n F  n (I n −W n ).Denote W ∗ n =  −1/2 n F  n W n F n  1/2 n which is a n ∗ × n ∗ matrix. This matrix can be regarded as a spatial weights matrix for the following transformed equation: Y ∗ nt = ␭ 0 W ∗ n Y ∗ nt + ␥ 0 Y ∗ n,t−1 + ␳ 0 W ∗ n Y ∗ n,t−1 + X ∗ nt ␤ 0 + c ∗ n0 + V ∗ nt , (14.29) where Y ∗ nt =  −1/2 n F  n (I n −W n )Y nt and other variables are defined correspond- ingly. Note that this transformed Y ∗ nt is an n ∗ dimensional vector. Hence, after the transformation, the observations at time period t have only n ∗ degrees of freedom. Equation 14.29 shall provide the structural parameters for esti- mation. This equation is in the format of a typical SAR model in panel data, where the number of observations is n ∗ T. A.4 Determinant and Inverse of S ∗ n (␭) ≡ I n ∗ − ␭W ∗ n We note that S ∗ n =  −1/2 n F  n S n F n  1/2 n .Let␮be ascalar. Because(I n −W n )·H n = 0, [F n ,H n ]  (␮I n − W n )[F n ,H n ] =  ␮I n ∗ − F  n W n F n −F  n W n H n −H  n W n F n ␮I n−n ∗ − H  n W n H n  =  ␮I n ∗ − F  n W n F n −F  n W n H n 0 (␮ −1)I n−n ∗  . Hence, |␮I n −W n |=(␮−1) n−n ∗ |␮I n ∗ −F  n W n F n |. Because |␮I n ∗ −W ∗ n |=|␮I n ∗ −  −1/2 n F  n W n F n  1/2 n |=|␮I n ∗ − F  n W n F n |, |␮I n − W n |=(␮ − 1) n−n ∗ |␮I n ∗ − W ∗ n |. As W n has (n − n ∗ ) unit eigenvalues, the eigenvalues of W ∗ n are exactly the remaining eigenvalues of W n , which are less than unity in the absolute value. Furthermore, |S ∗ n (␭)|= 1 (1 −␭) n−n ∗ |S n (␭)|. (14.30) Thus, the tractability in computing the determinant of S ∗ n (␭)isexactly that of S n (␭).WhenW n isconstructedas aweightsmatrix thatisrow-normalizedfrom an original symmetric matrix, Ord (1975) has suggested a computationally tractable method for the evaluation of |S n (␭)|at various ␭ for the ML method. This is useful for evaluating the determinant of S ∗ n (␭) even though the row sums of W ∗ n may not even be unity. Furthermore, a SAR model is an equilibrium model in the sense that the observed outcomes are determined by the equation. That is, the matrix S ∗ n (␭) shall be invertible. For the transformed equation (Equation 14.29), S ∗ n (␭)is invertible as long as the original matrices S n (␭)inEquation 14.1 is invertible. We can see that S ∗−1 n (␭) =  −1/2 n F  n S −1 n (␭)F n  1/2 n , (14.31) P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 422 Handbook of Empirical Economics and Finance because S ∗ n (␭) ·  −1/2 n F  n S −1 n (␭)F n  1/2 n =  −1/2 n F  n S n (␭)F n F  n S −1 n (␭)F n  1/2 n =  −1/2 n F  n S n (␭)(I n − H n H  n )S −1 n (␭)F n  1/2 n = I n ∗ −  −1/2 n F  n S n (␭)H n H  n S −1 n (␭)F n  1/2 n = I n ∗ , as H  n W n = H  n , H  n S −1 n (␭) = 1 1−␭ H  n and H  n F n = 0. A.5 About tr(G ∗ n (␭)) We have G ∗ n (␭) =  −1/2 n F  n G n (␭)F n  1/2 n .Thisis sobecause,fromEquation14.31, G ∗ n (␭) = W ∗ n S −1∗ n (␭) =  −1/2 n F  n W n F n F  n S −1 n (␭)F n  1/2 n =  −1/2 n F  n W n (I n − H n H  n )S −1 n (␭)F n  1/2 n =  −1/2 n F  n W n S −1 n (␭)F n  1/2 n −  −1/2 n F  n W n H n H  n S −1 n (␭)F n  1/2 n =  −1/2 n F  n W n S −1 n (␭)F n  1/2 n =  −1/2 n F  n G n (␭)F n  1/2 n , because H  n S −1 n (␭)F n = 1 1−␭ H  n F n = 0. Hence, tr(G ∗ n (␭)) = tr(F  n G n (␭)F n ) = tr[G n (␭)(I n − H n H  n )] = tr(G n (␭)) − n −n ∗ 1 − ␭ , (14.32) where the last equality holds because H  n W n = H  n and H  n S −1 n (␭) = 1 1−␭ H  n implies that tr(G n (␭)H n H  n ) = tr(H  n G n (␭)H n ) = tr(H  n W n S −1 n (␭)H n ) = 1 1 − ␭ tr(H  n H n ) = n −n ∗ 1 − ␭ . As G ∗2 n (␭) =  −1/2 n F  n G n (␭)F n F  n G n (␭)F n  1/2 n ,wehave tr(G ∗2 n (␭)) = tr(F  n G n (␭)F n F  n G n (␭)F n ) = tr(G n (␭)F n F  n G n (␭)F n F  n ) = tr(G n (␭)(I n − H n H  n )G n (␭)(I n − H n H  n )). Using H  n G n (␭) = 1 (1−␭) H  n and H  n H n = I n−n ∗ ,wehave [G n (␭)(I n − H n H  n )] 2 = [G n (␭)] 2 [I n − H n H  n ] and tr(G ∗2 n (␭)) = tr(G 2 n (␭)) − n −n ∗ (1 −␭) 2 , (14.33) because H  n G 2 n (␭)H n = 1 (1−␭) 2 H  n H n = 1 (1−␭) 2 I n−n ∗ .Interms of the eigenval- ues of W n ,asW n = R n ϖR −1 n , tr(G ∗ n (␭)) =  n j=m n +1 ϖ nj 1−␭ϖ nj and tr(G ∗2 n (␭)) =  n j=m n +1 ϖ 2 nj (1−␭ϖ nj ) 2 . P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 A Unified Estimation Approach for Spatial Dynamic Panel Data Models 423 Also, as J ∗ n = (I n −W n )   + n (I n −W n ) and (I n −W n )G n (␭) = G n (␭)(I n −W n ), Equation 14.32 implies that tr(J ∗ n G n (␭)) = tr(G n (␭)(I n − W n )(I n − W n )  F n  −1 n F  n ) = tr(G n (␭)F n F  n ) = tr(G n (␭)(I n − H n H  n )) = tr(G ∗ n (␭)). (14.34) For J ∗ n ,wehave tr(J ∗ n ) = tr((I n −W n )  F n  −1 n F  n (I n −W n )) = tr( −1 n  n ) = n ∗ by using Equation 14.28. The J ∗ n is an orthogonal projector. This is so, because J ∗ n is symmetric and J ∗ n J ∗ n = (I n −W n )   + n (I n −W n ) ·(I n −W n )   + n (I n −W n ) = (I n − W n )   + n  n  + n (I n − W n ) = (I n − W n )   + n (I n − W n ) = J ∗ n . B Lemmas for Some Statistics in the Model The following lemmas can be found in Yu, de Jong, and Lee (2008). These lemmas provide orders for relevant terms in the score and the Hessian matrix ofthelog-likelihoodfunction.They include also aCLT for linear andquadratic forms of disturbances. Denote U nt =  ∞ h=1 P nh V n,t+1−h , where {P nh } ∞ h=1 is a sequence of n ×n nonstochastic square matrices. Assumption A1 The disturbances {v it }, i = 1, 2, ,nand t = 1, 2, ,T,are i.i.d. across i and t with zero mean, variance ␴ 2 0 and E|v it | 4+␩ < ∞ for some ␩ > 0. Assumption A2  ∞ h=1 abs(P nh )isUB. Assumption A3 The elements of n × 1 vector D nt are nonstochastic and bounded, uniformly in n and t. Assumption A4 n is a nondecreasing function of T and T goes to infinity. Lemma 14.1 Under Assumptions A1 and A4, for an n × n nonstochastic matrix B n , uniformly bounded in row and column sums, 1 nT T  t=1 V  nt B n V nt − E( 1 nT T  t=1 V  nt B n V nt ) = O p  1 √ nT  , (14.35) 1 n ¯ V  nT B n ¯ V nT − E( 1 n ¯ V  nT B n ¯ V nT ) = O p  1 √ nT 2  , (14.36) and 1 nT T  t=1 ˜ V  nt B n ˜ V nt − E( 1 nT T  t=1 ˜ V  nt B n ˜ V nt ) = O p  1 √ nT  , (14.37) where E( 1 nT  T t=1 V  nt B n V nt ) = O(1), E( 1 n ¯ V  nT B n ¯ V nT ) = O(T −1 ) and E( 1 nT  T t=1 ˜ V  nt B n ˜ V nt ) = O(1). P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 424 Handbook of Empirical Economics and Finance Lemma 14.2 Under Assumptions A1, A2, and A4,  T n ( ¯ U  nT,−1 ¯ V nT − E( ¯ U  nT,−1 ¯ V nT )) = O p  1 √ T  , (14.38) where  T n E( ¯ U  nT,−1 ¯ V nT ) =  n T 1 n ␴ 2 0 tr   ∞ h=1 P nh  + O   n T 3  . For the lemma that follows, we will consider the following form: Q nT = T  t=1 (U  n,t−1 V nt + D  nt V nt + V  nt B n V nt − ␴ 2 0 tr(B n )) = T  t=1 n  i=1 z nt,i , where B n is a n × n nonstochastic symmetric matrix which is UB, and z nt,i = (u i,t−1 + d nti )v it + b n,ii (v 2 it − ␴ 2 0 ) + 2(  i−1 j=1 b n,i j v jt )v it , where b n,i j is the (i, j) elementofB n andd nti istheithelement of D nt .Then,forthemeanandvariance of Q nT , ␮ Q nT = 0 and ␴ 2 Q nT = T␴ 4 0 tr  ∞  h=1 P  nh P nh  + ␴ 2 0 T  t=1 D  nt D nt +T   ␮ 4 − 3␴ 4 0  n  i=1 b 2 n,ii + 2␴ 4 0 tr(B 2 n )  + 2␮ 3 T  t=1 n  i=1 d nti b n,ii , where ␮ s = Ev s it for s = 3, 4. Lemma 14.3 Under Assumptions A1, A2, A3, A4, and that B n is UB, if the sequence 1 nT ␴ 2 Q nT is bounded away from zero, then, Q nT ␴ Q nT d → N(0, 1). Denote Z nt = (Y n,t−1 ,W n Y n,t−1 ,X nt ), we are going to provide some lemmas related to (I n −W n ) ˜ Z nt ,(I n −W n ) ¯ Z nT and ˜ V nt , ¯ V nT of the model Equation 14.1. Lemma 14.4 Under Assumptions 1–7, for an n ×n nonstochastic UB matrix B n , 1 nT T  t=1 ˜ Z  nt (I n − W n )  B n (I n − W n ) ˜ Z nt − E 1 nT T  t=1 ˜ Z  nt (I n − W n )  B n (I n − W n ) ˜ Z nt = O p  1 √ nT  , (14.39) [...]... 440 17:3 C7035 C7035˙C015 Handbook of Empirical Economics and Finance tests The first one tests for the absence of random individual effects allowing for the possible presence of spatial lag dependence The second one tests for the absence of spatial lag dependence allowing for the possible presence of random individual effects As an alternative to the MLE, generalized method of moments have been proposed... 438 Handbook of Empirical Economics and Finance ␯t = (␯t1 , , ␯t N ), where ␯ti is assumed to be IIN(0, ␴2 ) and also independent ␯ of ␮i One can rewrite ⑀t as ⑀t = ( I N − ␳W) −1 ␯t = B −1 ␯t (15.4) where B = I N − ␳W and I N is an identity matrix of dimension N The model can be rewritten in matrix notation as y = X␤ + u (15.5) where y is now of dimension (NT × 1), X is (NT × k), ␤ is (k × 1) and. .. (14.42) P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 426 Handbook of Empirical Economics and Finance C.2 FOC and SOC of the Concentrated Log-Likelihood ∗ + ∗ ∗−1 Denote J n = ( In − Wn ) n ( In − Wn ) and G ∗ = Wn Sn By using tr G n (␭) − n n−n∗ n−n∗ ∗ 2 ∗2 tr (G n (␭)) = 1−␭ and tr (G n (␭)) − tr (G n (␭)) = (1−␭)2 (see Appendix A.5), the first-order derivatives of Equation 14.10 are... and Theorems D.1 Proof of nonsingularity of the information matrix The result can be proved by using an argument by contradiction For ␪0 ≡ limT→∞ ␪0 ,nT , where ␪0 ,nT is Equation 14.12, we shall prove that ␪0 ␣ = 0 P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 428 Handbook of Empirical Economics and Finance implies ␣ = 0, where ␣ = (␣1 , ␣2 , ␣3 ) , ␣2 , ␣3 are scalars and ␣1 is (k + 2)... Estate Finance and Economics 17(1):99–121 Kelejian, H H., and I R Prucha 2001 On the asymptotic distribution of the Moran I test statistic with applications Journal of Econometrics 104:219–257 Keller, W., and C H Shiue 2007 The origin of spatial interaction Journal of Econometrics 140:304–332 Korniotis, G M 2005 A dynamic panel estimator with both fixed and spatial effects Manuscript, Department of Finance, ... C7035 C7035˙C015 Handbook of Empirical Economics and Finance and heterogeneity using panel data Recent spatial panel data applications in economics include household level survey data from villages observed over time to study nutrition (see Case 1991); per capita expenditures on police to study their effect on reducing crime across counties (see Kelejian and Robinson 1992); the productivity of public capital... Pfaffermayr (2008a) suggest a generalized spatial panel model which encompasses the Anselin (1988) and Kapoor, Kelejian, and Prucha (2007) models and allows for spatial correlation in the P1: GOPAL JOSHI November 3, 2010 442 17:3 C7035 C7035˙C015 Handbook of Empirical Economics and Finance individual and remainder error components that may have different spatial autoregressive parameters They derive... the ( N × 1) vector of random individual effects u1 ␫T is a (T × 1) vector of ones and I N is an identity matrix of dimension N The vector of individual effects ␮ is assumed to be i.i.d N(0, ␴2 I N ), while the (n × 1) vector of remainder ␮ disturbances ␯ is assumed to be i.i.d N(0, ␴2 In ) Furthermore, the elements ␯ of ␮ and ␯ are assumed to be independent of each other Both u1 and u2 are spatially... − ␳u and ⑀ = u − ␳u, substituting these expressions in the six ¯ ¯ ¯ ¯ moment conditions we obtain a system of six equations involving the second moments of u, u and u Under the random effects specification considered, the ¯ ¯ OLS estimator of ␤ is consistent Using ␤ OL S one gets a consistent estimator of the disturbances u = y − X␤ OL S The GM estimator of ␴2 , ␴2 and ␳ is the ␯ 1 solution of the... Stock, and M W Watson 1990 Inference in linear time series models with some unit roots Econometrica 58:113–144 Su, L., and Z Yang 2007 QML estimation of dynamic panel data models with spatial errors Manuscript, Singapore Management University Theil, H 1971 Principles of Econometrics New York: John Wiley & Sons P1: NARESH CHANDRA November 12, 2010 434 18:3 C7035 C7035˙C014 Handbook of Empirical Economics . order 10 11 for the column of ␴ 2 , and 10 5 for other columns. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 416 Handbook of Empirical Economics and Finance 0.5 0.6 0.7 0.8 0.9. W n )G n l n = 1 1−␭ 0 (I n − W n )l n = 0. P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 426 Handbook of Empirical Economics and Finance C.2 FOC and SOC of the Concentrated Log-Likelihood Denote. given that  ␪ 0 ,nT is nonsingular and its inverse is P1: NARESH CHANDRA November 12, 2010 18:3 C7035 C7035˙C014 432 Handbook of Empirical Economics and Finance of order O(1), we have √ n ∗ T( ˆ ␪ nT −