Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation
Abstract
:1. Introduction
2. Model and Tests
2.1. LM and CD Tests
2.2. Assumptions and the Modified CD Test Statistic
3. Asymptotics
3.1. Asymptotic Distribution under the Null
3.2. Local Power Properties
4. Monte Carlo Simulations
4.1. Experimental Design
4.2. Simulation Results
5. Conclusions
Author Contributions
Conflicts of Interest
Appendix
Appendix A. Some Useful Lemmas
- (a)
- E;
- (b)
- E tr
- (c)
- E trtrtr
- (d)
- tr tr
- (a)
- E
- (b)
- E
- (c)
- For any E
- (a)
- tr
- (b)
- tr
- (c)
- tr for
Appendix B. Proof of the Theorems
Appendix B.1. Proof of Theorem 1
- (1)
- are mutually different. E
- (2)
- . By using Lemma A2, we have E
- (3)
- Since and are independent, we have
Appendix B.2. Proof of Theorem 2
- (1)
- .
- (2)
- (3)
Appendix B.3. Proof of Theorem 3
- (1)
- and are mutually different.
- (2)
- and
- (3)
Appendix B.4. Proof of Theorem 4
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- 1.The inclusion of predetermined variables, which is the weakly-exogenous case, alters the results.
- 2.We only consider the case that the number of non-zero factor loading vectors is N or of order N, which means the model has strong error cross-sectional correlation. For the weak error cross-sectional correlation case, we conjecture that it is similar to Pesaran [15].
Tests | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 5.75 | 5.90 | 5.50 | 4.75 | 6.45 | 5.90 | 4.80 | 5.55 | 5.15 | 6.45 | |
20 | 3.85 | 4.55 | 5.05 | 4.70 | 5.15 | 4.60 | 4.50 | 4.50 | 5.85 | 5.40 | |
30 | 4.45 | 4.10 | 4.70 | 5.10 | 4.60 | 4.40 | 4.80 | 4.45 | 4.50 | 6.25 | |
50 | 4.45 | 4.75 | 5.40 | 5.25 | 4.50 | 4.10 | 3.65 | 4.75 | 4.05 | 4.60 | |
100 | 4.65 | 4.85 | 4.20 | 5.65 | 5.30 | 4.35 | 4.80 | 4.70 | 4.35 | 4.95 | |
200 | 4.05 | 4.65 | 3.90 | 4.60 | 5.00 | 5.65 | 5.05 | 4.85 | 4.65 | 5.40 | |
10 | 5.60 | 5.50 | 5.25 | 4.10 | 6.00 | 5.60 | 4.70 | 5.05 | 4.70 | 5.65 | |
20 | 4.05 | 4.75 | 5.05 | 4.90 | 5.30 | 4.90 | 4.70 | 4.65 | 5.85 | 5.30 | |
30 | 4.90 | 4.45 | 4.85 | 5.20 | 5.00 | 5.20 | 5.20 | 4.55 | 5.00 | 6.05 | |
50 | 4.95 | 5.20 | 5.60 | 5.55 | 4.45 | 5.00 | 4.15 | 5.00 | 4.55 | 4.70 | |
100 | 5.65 | 5.15 | 4.50 | 5.95 | 5.45 | 5.15 | 5.65 | 5.05 | 4.50 | 5.05 | |
200 | 5.00 | 5.00 | 4.45 | 4.85 | 5.15 | 6.35 | 5.75 | 5.15 | 4.70 | 5.55 | |
10 | 6.75 | 6.05 | 6.10 | 6.00 | 5.60 | 6.60 | 6.85 | 7.65 | 7.95 | 6.60 | |
20 | 6.20 | 5.45 | 6.75 | 7.00 | 5.50 | 7.05 | 6.40 | 6.40 | 7.15 | 5.60 | |
30 | 6.20 | 6.25 | 5.40 | 6.35 | 5.95 | 7.65 | 5.95 | 6.35 | 5.85 | 7.00 | |
50 | 6.55 | 4.95 | 5.25 | 5.60 | 5.40 | 7.00 | 6.85 | 7.20 | 5.40 | 5.85 | |
100 | 8.10 | 5.45 | 5.40 | 4.60 | 4.55 | 7.00 | 5.85 | 6.10 | 5.85 | 5.90 | |
200 | 8.60 | 5.75 | 6.50 | 5.90 | 5.35 | 8.00 | 7.20 | 6.30 | 6.40 | 6.70 |
Tests | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 6.10 | 6.25 | 4.45 | 5.35 | 6.25 | 6.30 | 5.40 | 5.90 | 5.85 | 6.50 | |
20 | 5.15 | 4.80 | 5.05 | 4.60 | 5.30 | 5.20 | 5.35 | 4.70 | 6.15 | 4.75 | |
30 | 4.50 | 4.35 | 4.20 | 5.35 | 4.95 | 5.55 | 4.75 | 4.90 | 5.30 | 6.15 | |
50 | 5.25 | 4.50 | 5.30 | 5.70 | 4.30 | 5.00 | 4.65 | 4.60 | 4.35 | 4.85 | |
100 | 4.75 | 5.35 | 4.50 | 5.45 | 5.60 | 5.80 | 4.15 | 5.45 | 4.35 | 4.90 | |
200 | 4.35 | 4.95 | 3.50 | 4.50 | 4.90 | 6.20 | 6.30 | 4.30 | 4.30 | 5.50 | |
10 | 7.60 | 9.35 | 8.40 | 10.05 | 11.10 | 7.80 | 7.75 | 10.30 | 10.25 | 10.95 | |
20 | 6.60 | 8.30 | 9.95 | 9.10 | 10.90 | 7.00 | 8.95 | 9.30 | 10.70 | 10.50 | |
30 | 6.45 | 8.35 | 8.30 | 10.50 | 10.60 | 7.90 | 9.65 | 9.50 | 10.80 | 10.60 | |
50 | 7.45 | 7.95 | 10.75 | 11.30 | 9.65 | 7.55 | 7.90 | 9.20 | 9.70 | 9.15 | |
100 | 6.50 | 9.35 | 9.00 | 10.85 | 11.55 | 7.85 | 8.35 | 10.60 | 9.30 | 10.20 | |
200 | 6.65 | 8.45 | 8.45 | 9.70 | 10.95 | 9.90 | 9.50 | 9.35 | 9.65 | 11.20 | |
10 | 37.95 | 54.40 | 57.10 | 59.55 | 60.70 | 39.15 | 53.00 | 56.50 | 60.75 | 61.55 | |
20 | 81.55 | 96.00 | 96.80 | 98.25 | 97.90 | 83.25 | 95.45 | 97.05 | 97.70 | 98.20 | |
30 | 98.30 | 100.00 | 100.00 | 100.00 | 100.00 | 98.45 | 100.00 | 100.00 | 100.00 | 100.00 | |
50 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
100 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
200 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |
Tests | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 6.10 | 6.25 | 4.90 | 6.15 | 6.75 | 6.05 | 4.80 | 6.10 | 6.00 | 5.65 | |
20 | 4.75 | 5.65 | 4.65 | 4.70 | 5.00 | 4.85 | 5.60 | 4.50 | 5.55 | 4.80 | |
30 | 4.15 | 4.85 | 4.00 | 4.55 | 4.65 | 5.50 | 4.25 | 5.75 | 5.10 | 6.65 | |
50 | 4.15 | 4.50 | 5.20 | 5.45 | 4.40 | 5.25 | 5.35 | 4.60 | 4.40 | 4.35 | |
100 | 4.35 | 4.80 | 4.80 | 5.45 | 4.80 | 5.75 | 4.15 | 5.30 | 4.05 | 5.10 | |
200 | 4.85 | 4.60 | 4.05 | 4.55 | 5.05 | 7.80 | 5.35 | 4.95 | 4.20 | 4.55 | |
10 | 6.80 | 9.65 | 10.20 | 14.55 | 16.80 | 6.55 | 8.25 | 12.25 | 13.90 | 16.30 | |
20 | 5.75 | 9.50 | 11.35 | 13.25 | 16.85 | 5.90 | 9.60 | 11.50 | 15.05 | 15.45 | |
30 | 5.65 | 9.80 | 10.00 | 13.30 | 14.05 | 7.35 | 9.65 | 12.00 | 15.20 | 17.15 | |
50 | 5.90 | 8.45 | 11.95 | 14.80 | 14.10 | 7.10 | 9.55 | 9.70 | 12.40 | 15.80 | |
100 | 6.05 | 10.00 | 10.40 | 14.70 | 16.55 | 7.25 | 8.70 | 12.25 | 13.85 | 15.00 | |
200 | 6.65 | 9.00 | 10.25 | 13.30 | 16.70 | 9.40 | 10.3 | 10.85 | 13.70 | 16.10 | |
10 | 37.95 | 54.40 | 57.10 | 59.55 | 60.70 | 27.60 | 66.30 | 82.45 | 90.80 | 95.35 | |
20 | 55.50 | 97.90 | 99.85 | 100.00 | 100.00 | 59.95 | 98.40 | 99.85 | 100.00 | 100.00 | |
30 | 98.30 | 99.95 | 100.00 | 100.00 | 100.00 | 82.75 | 100.00 | 100.00 | 100.00 | 100.00 | |
50 | 97.80 | 100.00 | 100.00 | 100.00 | 100.00 | 98.60 | 100.00 | 100.00 | 100.00 | 100.00 | |
100 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
200 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |
Tests | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 6.95 | 6.45 | 4.90 | 6.20 | 5.85 | 7.20 | 5.25 | 6.40 | 5.40 | 5.45 | |
20 | 5.40 | 5.55 | 4.95 | 4.75 | 4.95 | 6.40 | 5.70 | 4.95 | 5.55 | 4.70 | |
30 | 4.65 | 4.75 | 4.05 | 4.80 | 4.65 | 7.45 | 4.60 | 5.95 | 5.10 | 6.50 | |
50 | 4.95 | 4.95 | 5.25 | 5.30 | 4.50 | 7.50 | 5.70 | 4.80 | 4.35 | 4.80 | |
100 | 5.05 | 5.15 | 4.60 | 5.10 | 4.90 | 10.25 | 5.10 | 4.65 | 4.00 | 4.80 | |
200 | 5.75 | 4.65 | 4.45 | 4.85 | 5.20 | 17.45 | 6.60 | 5.75 | 4.50 | 4.25 | |
10 | 9.10 | 15.95 | 16.35 | 22.50 | 24.30 | 10.95 | 13.80 | 19.20 | 21.70 | 25.15 | |
20 | 8.30 | 14.40 | 17.80 | 20.15 | 25.05 | 10.10 | 14.80 | 18.90 | 22.85 | 23.15 | |
30 | 8.30 | 15.40 | 17.70 | 21.55 | 22.55 | 10.95 | 15.25 | 19.25 | 23.55 | 24.25 | |
50 | 8.70 | 14.85 | 18.80 | 22.70 | 23.40 | 11.75 | 15.40 | 17.30 | 19.15 | 23.95 | |
100 | 9.35 | 15.90 | 17.50 | 22.15 | 24.20 | 17.20 | 14.45 | 17.95 | 22.05 | 22.70 | |
200 | 9.50 | 14.05 | 18.35 | 20.00 | 24.95 | 25.45 | 17.00 | 18.55 | 21.35 | 24.65 | |
10 | 83.65 | 98.45 | 99.45 | 99.75 | 99.80 | 83.65 | 98.40 | 99.70 | 99.90 | 100.00 | |
20 | 99.85 | 100.00 | 100.00 | 100.00 | 100.00 | 99.85 | 100.00 | 100.00 | 100.00 | 100.00 | |
30 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
50 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
100 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |
200 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |
DGP | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 14.55 | 23.95 | 30.30 | 45.40 | 63.05 | 21.95 | 30.75 | 33.65 | 46.00 | 66.10 | |
20 | 35.70 | 56.65 | 68.95 | 84.05 | 95.95 | 47.30 | 63.25 | 75.80 | 86.00 | 97.40 | |
30 | 59.65 | 81.70 | 91.75 | 97.65 | 99.95 | 69.75 | 87.50 | 92.60 | 98.00 | 99.95 | |
50 | 83.65 | 96.60 | 99.30 | 100.00 | 100.00 | 88.75 | 98.00 | 99.55 | 100.00 | 100.00 | |
100 | 96.75 | 99.95 | 100.00 | 100.00 | 100.00 | 98.90 | 99.90 | 100.00 | 100.00 | 100.00 | |
200 | 99.70 | 100.00 | 100.00 | 100.00 | 100.00 | 99.70 | 100.00 | 100.00 | 100.00 | 100.00 | |
10 | 18.95 | 23.95 | 32.40 | 38.10 | 56.75 | 26.95 | 35.00 | 28.90 | 37.15 | 61.25 | |
20 | 45.60 | 62.10 | 69.95 | 81.45 | 94.20 | 55.10 | 67.45 | 74.85 | 85.65 | 96.60 | |
30 | 68.80 | 83.50 | 92.30 | 97.60 | 99.75 | 78.15 | 90.85 | 92.70 | 97.40 | 99.85 | |
50 | 88.55 | 97.45 | 99.40 | 100.00 | 100.00 | 92.90 | 98.50 | 99.65 | 100.00 | 100.00 | |
100 | 98.80 | 100.00 | 100.00 | 100.00 | 100.00 | 99.60 | 99.95 | 100.00 | 100.00 | 100.00 | |
200 | 99.90 | 100.00 | 100.00 | 100.00 | 100.00 | 99.85 | 100.00 | 100.00 | 100.00 | 100.00 | |
10 | 7.70 | 7.70 | 10.00 | 10.80 | 14.80 | 9.65 | 10.35 | 8.80 | 9.60 | 19.60 | |
20 | 22.05 | 18.85 | 24.25 | 27.80 | 39.50 | 24.85 | 22.35 | 23.40 | 30.60 | 46.20 | |
30 | 37.75 | 37.45 | 46.15 | 48.90 | 75.00 | 41.75 | 47.35 | 44.15 | 53.15 | 71.25 | |
50 | 66.50 | 66.75 | 71.60 | 83.10 | 96.20 | 66.25 | 72.35 | 82.45 | 88.20 | 98.00 | |
100 | 91.15 | 96.60 | 98.75 | 99.90 | 100.00 | 90.45 | 98.55 | 99.40 | 99.95 | 100.00 | |
200 | 98.95 | 100.00 | 100.00 | 100.00 | 100.00 | 98.45 | 99.95 | 100.00 | 100.00 | 100.00 |
DGP | (N,T) | Normal | Chi-Squared | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 20 | 30 | 50 | 100 | 10 | 20 | 30 | 50 | 100 | ||
10 | 38.85 | 60.55 | 72.20 | 88.25 | 97.30 | 43.05 | 67.15 | 72.55 | 88.45 | 97.70 | |
20 | 37.45 | 61.70 | 76.00 | 92.15 | 99.05 | 39.25 | 61.25 | 76.80 | 89.55 | 99.10 | |
30 | 39.60 | 64.55 | 78.60 | 92.00 | 99.60 | 40.30 | 65.65 | 78.80 | 91.90 | 99.35 | |
50 | 40.05 | 66.45 | 79.15 | 92.70 | 99.75 | 39.95 | 66.55 | 78.65 | 94.65 | 99.70 | |
100 | 33.60 | 62.70 | 80.55 | 92.55 | 99.65 | 37.85 | 64.65 | 79.20 | 94.40 | 99.90 | |
200 | 40.65 | 64.50 | 80.65 | 94.70 | 99.8 | 37.75 | 62.50 | 81.25 | 95.65 | 99.80 | |
10 | 37.20 | 53.95 | 68.20 | 79.20 | 92.10 | 42.85 | 63.20 | 61.15 | 78.00 | 94.80 | |
20 | 38.25 | 56.50 | 69.30 | 82.90 | 95.85 | 38.55 | 55.50 | 68.65 | 83.70 | 97.20 | |
30 | 37.90 | 56.90 | 71.80 | 84.65 | 98.10 | 38.70 | 62.00 | 66.25 | 85.70 | 96.90 | |
50 | 38.80 | 59.80 | 71.40 | 86.60 | 98.60 | 39.70 | 59.15 | 71.25 | 89.00 | 99.00 | |
100 | 38.85 | 57.85 | 70.90 | 86.60 | 98.75 | 35.25 | 59.85 | 72.55 | 88.95 | 98.60 | |
200 | 40.75 | 55.95 | 74.40 | 87.75 | 98.80 | 33.80 | 56.00 | 70.85 | 90.40 | 99.10 | |
10 | 29.00 | 43.40 | 58.05 | 70.20 | 85.90 | 32.75 | 49.75 | 51.30 | 67.40 | 88.20 | |
20 | 31.05 | 43.55 | 56.65 | 72.10 | 89.10 | 28.35 | 43.45 | 54.80 | 71.35 | 91.35 | |
30 | 30.00 | 45.70 | 59.35 | 71.35 | 94.20 | 28.10 | 48.10 | 54.00 | 73.05 | 91.90 | |
50 | 33.05 | 45.30 | 54.40 | 71.70 | 93.30 | 27.30 | 43.90 | 58.00 | 75.75 | 94.45 | |
100 | 30.60 | 45.15 | 55.50 | 75.40 | 94.95 | 21.80 | 45.45 | 57.85 | 77.35 | 94.75 | |
200 | 30.30 | 42.05 | 58.15 | 75.75 | 95.15 | 21.05 | 38.80 | 55.70 | 77.50 | 95.80 |
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Baltagi, B.H.; Kao, C.; Peng, B. Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation. Econometrics 2016, 4, 44. https://doi.org/10.3390/econometrics4040044
Baltagi BH, Kao C, Peng B. Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation. Econometrics. 2016; 4(4):44. https://doi.org/10.3390/econometrics4040044
Chicago/Turabian StyleBaltagi, Badi H., Chihwa Kao, and Bin Peng. 2016. "Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation" Econometrics 4, no. 4: 44. https://doi.org/10.3390/econometrics4040044
APA StyleBaltagi, B. H., Kao, C., & Peng, B. (2016). Testing Cross-Sectional Correlation in Large Panel Data Models with Serial Correlation. Econometrics, 4(4), 44. https://doi.org/10.3390/econometrics4040044