We can see that, when0+ 0+ 0< 1, the test is more powerful by using the corresponding method without any transformation; when0+ 0+ 0 > 1, the power curves are irregular and we need to re
Trang 1CP 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
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
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):
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
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
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 1011for the column of 2 , and
10 5 for other columns.
Trang 20.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
sum α=5%
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 H0 : 0 + 0 + 0 = 1.
unified approach to get the power curves The results are in Figure 14.1 Forthe two-sided tests, the sum0 + 0+ 0 under the alternative hypothesisranges from 0.65 to 1.35 with a 2000.7 increment; for the one-sided test with
H1:0+ 0+ 0 < 1, the sum 0+ 0+ 0ranges from 0.65 to 1.0 with a 0.35
200increment From Figure 14.2, we can see that the empirical sizes18are close to
the theoretical ones and the tests are more powerful when T= 50 than those
for the small T = 10 The power seems reasonable for the large T = 50 We run
additional simulations where we use the corresponding estimation methodwithout any transformation Figure 14.2 is the counterparts19 of Table 14.1
18For 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.
Trang 30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
sum α=5%
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 H0 : 0 + 0 + 0 = 1.
We can see that, when0+ 0+ 0< 1, the test is more powerful by using the
corresponding method without any transformation; when0+ 0+ 0 > 1,
the power curves are irregular and we need to rely on the unified approachfor the inferences.20
14.5 Conclusion
This chapter establishes asymptotic properties of QMLEs for SDPD modelswith both time and individual fixed effects when both the number of individ-
uals n and the number of time periods T can be large Instead of using different
20For 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.
Trang 4estimation methods depending on whether the DGP has time effects or notand whether the DGP is stable or not, we propose a data transformation ap-proach to eliminate both the time effects and the possible unstable or explosiveeffects 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 timeseries, the co-integrating vector is known and does not need to be estimated.With the proposed data transformation, the possible unstable or explosivecomponents and time effects can be eliminated
The transformation uses the co-integrating matrix The effective sample size
n∗ after transformation corresponds to the co-integration rank, which is thenumber of eigenvalues not equal to the unity This transformation is of partic-ular value when the process may contain explosive roots, as usual estimationmethods 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 eliminatethe bias and yield a centered confidence interval Monte Carlo experimentshave demonstrated a desirable finite sample performance of the estimator Atest statistic for testing possible spatial co-integration is also considered InLee and Yu (2010b), this unified estimation approach is applied to study themarket integration in Keller and Shiue (2007) with the SDPD model and testfor the spatial co-integration
Appendices
A Some Notes
From Subsection 14.2.1, the eigenvalues matrix of A ncan be decomposed as
as-As|0| < 1 (implied by Assumptions 1 and 3), the derivative is zero if and
Trang 5only if0+ 00 = 0, i.e., 0 = −00 In this situation, d ni = 0 + 0 ni
1− 0 ni = 0,and all|d ni | < 1 if |0| < 1.21The d niis a strictly increasing function of niifand only if0+ 00> 0; otherwise it is a strictly decreasing function of ni
when0+ 00< 0 Let 0+ 0+ 0 = 1 + a, where a is a constant We have
the stable case when0+ 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
condition0+ 00 > 0 (< 0) is equivalent to (1 − 0)(1− 0)> −a (< −a)
because (1− 0)(1− 0)= 0+ 00− a.
Assume that d niis an increasing function of ni As W nis row-normalized,
−1 ≤ ni ≤ 1 for all i With the relation d ni = 0 + 0 ni
1− 0 ni on [−1, 1], dni = 0 − 0
1+ 0
at ni = −1, and d ni =0 + 0
1− 0 at ... cial feature of the transformed Equation 14.4 is that the variance matrix of
diago-That is, the columns of F nconsist of eigenvectors of nonzero eigenvalues and
those... to Equation A.14 in Yu,
de Jong, and Lee (2008)
D Proofs for Claims and Theorems
D.1 Proof of nonsingularity of the information matrix
The... The elements of n × vector D nt are nonstochastic and
bounded, uniformly in n and t.
Assumption A4n is a nondecreasing function of T and T goes