Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models
Abstract
:1. Introduction
2. Subset-Continuous-Updating GMM Estimator
3. Monte Carlo Experiments
- θ = {0.5, 0.8} for a moderate vs. high persistence in yit,
- ρ = {0.5, 0.8} for a moderate vs. high persistence in xit,
- λ = {−0.1, −0.4} for a low vs. high level of idiosynchratic-error endogeneity in xit,
- = {1/4, 4} for a small vs. large variance of fixed effects relative to idiosyncratic errors.
Estimation accuracy on θ | T = 4 | T = 8 | T = 12 |
SCUDIF outperforms DIF | 75.0% | 100.0% | 75.0% |
SCUSYS outperforms SYS | 87.5% | 100.0% | 100.0% |
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- M. Arellano, and S. Bond. “Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations.” Rev. Econ. Stud. 58 (1991): 277–297. [Google Scholar] [CrossRef]
- S.C. Ahn, and P. Schmidt. “Efficient estimation of models for dynamic panel data.” J. Econom. 68 (1995): 5–27. [Google Scholar] [CrossRef]
- R. Blundell, and S. Bond. “Initial conditions and moment restrictions in dynamic panel data models.” J. Econom. 87 (1998): 115–143. [Google Scholar] [CrossRef]
- M. Arellano, and O. Bover. “Another look at the instrumental variable estimation of error-component models.” J. Econom. 68 (1995): 29–51. [Google Scholar] [CrossRef]
- R. Blundell, S. Bond, and F. Windmeijer. “Estimation in Dynamic Panel Data Models: Improving on the Performance of the Standard GMM Estimator.” In Nonstationary Panels, Panel Cointegration and Dynamic Panels. Edited by B.H. Baltagi, T.B. Fomby and R.C. Hill. (Book Series: Advances in Econometrics); Bingley, UK: Emerald Group Publishing Limited, 2001, Volume 15, pp. 53–91. [Google Scholar]
- K. Hayakawa. “Small sample bias properties of the system GMM estimator in dynamic panel data models.” Econ. Lett. 95 (2007): 32–38. [Google Scholar] [CrossRef]
- M.J.G. Bun, and F. Windmeijer. “The weak instrument problem of the system GMM estimator in dynamic panel data models.” Econom. J. 13 (2010): 95–126. [Google Scholar] [CrossRef]
- W.K. Newey, and F. Windmeijer. “Generalized method of moments with many weak moment conditions.” Econometrica 77 (2009): 687–719. [Google Scholar]
- L.P. Hansen, J. Heaton, and A. Yaron. “Finite-sample properties of some alternative GMM estimators.” J. Bus. Econ. Stat. 14 (1996): 262–280. [Google Scholar]
- S.G. Donald, and W.K. Newey. “A Jackknife Interpretation of the Continuous Updating Estimator.” Econ. Lett. 67 (2000): 239–243. [Google Scholar] [CrossRef]
- J.D. Sargan. “The estimation of economic relationships using instrumental variables.” Econometrica 26 (1958): 393–415. [Google Scholar] [CrossRef]
- L.P. Hansen. “Large sample properties of generalized method of moments estimators.” Econometrica 50 (1982): 1029–1054. [Google Scholar] [CrossRef]
- G.E.P. Box, and G.M. Jenkins. Time Series Analysis: Forecasting and Control. San Francisco, CA, USA: Holden-Day, 1976. [Google Scholar]
- F. Windmeijer. “A finite sample correction for the variance of linear efficient two-step GMM estimators.” J. Econom. 126 (2005): 25–51. [Google Scholar] [CrossRef]
- 1.To obtain a consistent first-step estimator, we use
- 2.In the Monte Carlo experiments, we conduct a bounded optimization over using the two-step DIF or SYS GMM estimator of θ as the starting value until a convergence criterion is met. We set both the step tolerance and the function tolerance to a relatively small number, 10−8.
- 3.After we draw , , and from the normal distribution, we simply set the initial values of and to
- 4.The standard error estimates for the two-step DIF and SYS GMM estimators of θ and β are calculated as inWindmeijer [14]. The same is obtained for the SCUDIF and SCUSYS GMM estimators of β conditional on θ. The standard error estimates for the SCUDIF and SCUSYS GMM estimators of θ are instead obtained from the Hessian matrix of the single-dimensional optimization problem.
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
DIF | 0.0625 | 0.0914 | 0.0922 | 5.5 | 0.2341 | 0.3479 | 0.3463 | 5.4 | 5.2 | 0.1240 | 0.1707 | 0.1721 | 8.7 | 0.3543 | 0.5018 | 0.5057 | 7.2 | 5.7 | |
SCUDIF | 0.0627 | 0.0945 | 0.0665 | 16.7 | 0.2328 | 0.3538 | 0.3180 | 7.4 | 4.8 | 0.1287 | 0.2004 | 0.1447 | 25.8 | 0.3679 | 0.5760 | 0.3384 | 23.0 | 4.1 | |
SYS | 0.0366 | 0.0525 | 0.0525 | 5.4 | 0.1565 | 0.2321 | 0.2309 | 4.9 | 4.6 | 0.0652 | 0.0749 | 0.0753 | 17.4 | 0.2203 | 0.3025 | 0.3019 | 9.7 | 10.2 | |
SCUSYS | 0.0377 | 0.0546 | 0.0375 | 18.4 | 0.1569 | 0.2327 | 0.2282 | 5.3 | 4.4 | 0.0515 | 0.0792 | 0.0499 | 26.8 | 0.2191 | 0.3196 | 0.2978 | 9.5 | 6.0 | |
DIF | 0.0593 | 0.0872 | 0.0877 | 5.5 | 0.2811 | 0.4285 | 0.4309 | 4.5 | 4.9 | 0.1139 | 0.1598 | 0.1612 | 8.6 | 0.5146 | 0.7863 | 0.7799 | 6.9 | 6.2 | |
SCUDIF | 0.0592 | 0.0917 | 0.0642 | 16.9 | 0.2859 | 0.4456 | 0.3519 | 11.1 | 4.3 | 0.1316 | 0.2141 | 0.1482 | 34.0 | 0.6021 | 1.0479 | 0.4074 | 40.6 | 4.1 | |
SYS | 0.0335 | 0.0488 | 0.0485 | 6.0 | 0.1037 | 0.1592 | 0.1632 | 4.2 | 4.3 | 0.0568 | 0.0639 | 0.0612 | 20.0 | 0.1986 | 0.2217 | 0.2214 | 19.0 | 14.2 | |
SCUSYS | 0.0340 | 0.0509 | 0.0346 | 19.0 | 0.1042 | 0.1605 | 0.1605 | 4.7 | 4.1 | 0.0462 | 0.0767 | 0.0400 | 33.7 | 0.2124 | 0.2565 | 0.2249 | 22.2 | 10.3 | |
DIF | 0.1423 | 0.2010 | 0.2058 | 8.5 | 0.3243 | 0.4624 | 0.4726 | 7.2 | 6.2 | 0.3336 | 0.3647 | 0.3702 | 19.5 | 0.6584 | 0.7767 | 0.7656 | 17.5 | 7.2 | |
SCUDIF | 0.1442 | 0.1778 | 0.2230 | 14.9 | 0.2950 | 0.4431 | 0.3114 | 16.5 | 4.5 | 0.2000 | 0.3446 | 0.4377 | 26.5 | 0.4862 | 0.7555 | 0.3129 | 39.5 | 4.3 | |
SYS | 0.0388 | 0.0593 | 0.0578 | 7.2 | 0.1475 | 0.2223 | 0.2270 | 4.4 | 4.5 | 0.0836 | 0.0652 | 0.0605 | 48.0 | 0.1695 | 0.2529 | 0.2671 | 4.9 | 12.8 | |
SCUSYS | 0.0422 | 0.0683 | 0.0419 | 23.6 | 0.1508 | 0.2268 | 0.2213 | 5.5 | 4.1 | 0.0738 | 0.1466 | 0.0980 | 48.9 | 0.2066 | 0.3466 | 0.2877 | 9.0 | 5.7 | |
DIF | 0.1073 | 0.1592 | 0.1598 | 6.7 | 0.4205 | 0.6486 | 0.6479 | 6.0 | 5.4 | 0.2202 | 0.2791 | 0.2827 | 12.6 | 0.8422 | 1.1380 | 1.1334 | 11.2 | 4.9 | |
SCUDIF | 0.1124 | 0.1658 | 0.1786 | 18.1 | 0.4179 | 0.7014 | 0.3375 | 28.8 | 4.0 | 0.2000 | 0.3246 | 0.3684 | 32.6 | 0.7872 | 1.3599 | 0.3669 | 56.6 | 2.6 | |
SYS | 0.0301 | 0.0461 | 0.0459 | 6.4 | 0.0910 | 0.1433 | 0.1450 | 4.3 | 4.7 | 0.0586 | 0.0457 | 0.0429 | 44.3 | 0.1138 | 0.1752 | 0.1871 | 4.8 | 9.5 | |
SCUSYS | 0.0330 | 0.0539 | 0.0338 | 23.5 | 0.0935 | 0.1528 | 0.1400 | 6.0 | 4.1 | 0.0542 | 0.0893 | 0.0389 | 53.6 | 0.1523 | 0.2717 | 0.2051 | 13.4 | 5.5 | |
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
DIF | 0.0974 | 0.1288 | 0.1319 | 10.0 | 0.2623 | 0.3349 | 0.3407 | 12.3 | 7.8 | 0.1909 | 0.1961 | 0.2020 | 22.4 | 0.4626 | 0.4750 | 0.4872 | 22.8 | 11.6 | |
SCUDIF | 0.0936 | 0.1500 | 0.1037 | 18.2 | 0.2483 | 0.3835 | 0.2241 | 24.5 | 5.3 | 0.1776 | 0.2575 | 0.1893 | 34.2 | 0.4215 | 0.6052 | 0.2334 | 49.1 | 4.9 | |
SYS | 0.0452 | 0.0668 | 0.0666 | 6.1 | 0.1237 | 0.1712 | 0.1727 | 6.9 | 5.2 | 0.0921 | 0.1027 | 0.0981 | 23.5 | 0.1753 | 0.2489 | 0.2474 | 9.9 | 10.5 | |
SCUSYS | 0.0483 | 0.0718 | 0.0479 | 20.4 | 0.1235 | 0.1743 | 0.1574 | 9.9 | 4.9 | 0.0735 | 0.1242 | 0.0678 | 33.8 | 0.1864 | 0.2912 | 0.2082 | 16.0 | 4.2 | |
DIF | 0.0631 | 0.0888 | 0.0899 | 7.6 | 0.2501 | 0.3409 | 0.3472 | 8.6 | 6.6 | 0.1144 | 0.1404 | 0.1425 | 14.4 | 0.4260 | 0.5365 | 0.5426 | 14.0 | 7.7 | |
SCUDIF | 0.0626 | 0.0993 | 0.0682 | 17.9 | 0.2390 | 0.3787 | 0.2361 | 20.7 | 5.2 | 0.1116 | 0.1890 | 0.1283 | 33.9 | 0.4030 | 0.7097 | 0.2556 | 43.9 | 4.0 | |
SYS | 0.0331 | 0.0484 | 0.0484 | 5.5 | 0.0705 | 0.1057 | 0.1059 | 4.7 | 4.8 | 0.0611 | 0.0681 | 0.0639 | 22.8 | 0.1265 | 0.1637 | 0.1656 | 15.5 | 10.8 | |
SCUSYS | 0.0349 | 0.0513 | 0.0345 | 19.7 | 0.0713 | 0.1073 | 0.0997 | 6.6 | 4.6 | 0.0459 | 0.0837 | 0.0415 | 32.7 | 0.1371 | 0.2080 | 0.1618 | 15.8 | 5.5 | |
DIF | 0.2941 | 0.2778 | 0.2659 | 30.4 | 0.6348 | 0.5928 | 0.5655 | 31.6 | 12.9 | 0.5076 | 0.3054 | 0.2996 | 53.7 | 1.0708 | 0.6656 | 0.6335 | 53.0 | 10.3 | |
SCUDIF | 0.2000 | 0.2754 | 0.2961 | 21.6 | 0.3877 | 0.6099 | 0.2122 | 42.9 | 5.8 | 0.2216 | 0.3771 | 0.4255 | 33.1 | 0.6169 | 0.8182 | 0.1970 | 60.8 | 5.9 | |
SYS | 0.0503 | 0.0809 | 0.0782 | 6.9 | 0.1254 | 0.1861 | 0.1861 | 5.4 | 4.9 | 0.0966 | 0.0698 | 0.0643 | 54.1 | 0.1238 | 0.1931 | 0.2059 | 3.9 | 8.4 | |
SCUSYS | 0.0593 | 0.0950 | 0.0624 | 28.2 | 0.1312 | 0.1992 | 0.1567 | 13.3 | 4.0 | 0.1053 | 0.1853 | 0.1103 | 59.9 | 0.1845 | 0.3784 | 0.2153 | 15.8 | 3.3 | |
DIF | 0.1067 | 0.1350 | 0.1360 | 13.0 | 0.4377 | 0.5473 | 0.5444 | 13.9 | 8.5 | 0.2115 | 0.1941 | 0.1923 | 29.4 | 0.8441 | 0.7932 | 0.7753 | 28.7 | 9.2 | |
SCUDIF | 0.1034 | 0.1491 | 0.1360 | 21.7 | 0.3989 | 0.6164 | 0.2341 | 44.3 | 4.6 | 0.2000 | 0.2705 | 0.2288 | 38.1 | 0.6728 | 1.1154 | 0.2419 | 67.7 | 3.3 | |
SYS | 0.0272 | 0.0429 | 0.0428 | 5.9 | 0.0601 | 0.0940 | 0.0945 | 3.5 | 5.2 | 0.0540 | 0.0459 | 0.0430 | 42.0 | 0.0820 | 0.1300 | 0.1378 | 6.4 | 8.4 | |
SCUSYS | 0.0309 | 0.0499 | 0.0317 | 23.6 | 0.0619 | 0.1029 | 0.0837 | 8.1 | 4.6 | 0.0510 | 0.0943 | 0.0385 | 53.0 | 0.1167 | 0.2754 | 0.1502 | 16.4 | 4.1 |
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
DIF | 0.0251 | 0.0345 | 0.0347 | 7.1 | 0.0969 | 0.1304 | 0.1289 | 7.6 | 5.5 | 0.0353 | 0.0443 | 0.0443 | 8.9 | 0.1080 | 0.1381 | 0.1381 | 9.1 | 5.6 | |
SCUDIF | 0.0242 | 0.0361 | 0.0245 | 19.3 | 0.0943 | 0.1305 | 0.1263 | 7.7 | 5.2 | 0.0319 | 0.0479 | 0.0327 | 32.4 | 0.1011 | 0.1397 | 0.1288 | 9.2 | 5.0 | |
SYS | 0.0167 | 0.0244 | 0.0241 | 5.8 | 0.0778 | 0.1116 | 0.1101 | 6.1 | 5.4 | 0.0383 | 0.0349 | 0.0320 | 22.6 | 0.1174 | 0.1435 | 0.1340 | 12.7 | 11.0 | |
SCUSYS | 0.0171 | 0.0252 | 0.0171 | 19.7 | 0.0773 | 0.1117 | 0.1093 | 6.2 | 5.3 | 0.0197 | 0.0294 | 0.0205 | 29.4 | 0.1017 | 0.1421 | 0.1360 | 8.1 | 5.6 | |
DIF | 0.0235 | 0.0325 | 0.0328 | 7.0 | 0.0880 | 0.1203 | 0.1197 | 6.9 | 5.2 | 0.0319 | 0.0414 | 0.0415 | 8.5 | 0.1098 | 0.1444 | 0.1431 | 8.1 | 5.3 | |
SCUDIF | 0.0229 | 0.0343 | 0.0238 | 21.8 | 0.0837 | 0.1211 | 0.1111 | 8.1 | 4.7 | 0.0305 | 0.0460 | 0.0291 | 39.6 | 0.1009 | 0.1506 | 0.1167 | 13.0 | 4.7 | |
SYS | 0.0158 | 0.0228 | 0.0225 | 6.2 | 0.0512 | 0.0760 | 0.0751 | 5.7 | 4.8 | 0.0462 | 0.0300 | 0.0275 | 40.2 | 0.1493 | 0.0962 | 0.0909 | 41.9 | 17.1 | |
SCUSYS | 0.0161 | 0.0237 | 0.0162 | 21.7 | 0.0513 | 0.0763 | 0.0744 | 6.2 | 4.8 | 0.0200 | 0.0278 | 0.0178 | 39.7 | 0.1441 | 0.1135 | 0.1056 | 31.2 | 9.6 | |
DIF | 0.0568 | 0.0584 | 0.0595 | 13.8 | 0.1322 | 0.1465 | 0.1454 | 12.6 | 6.4 | 0.1023 | 0.0800 | 0.0820 | 22.8 | 0.1952 | 0.1738 | 0.1715 | 19.4 | 7.0 | |
SCUDIF | 0.0431 | 0.0650 | 0.0439 | 32.4 | 0.1069 | 0.1520 | 0.1246 | 12.3 | 5.0 | 0.0658 | 0.0994 | 0.0542 | 52.4 | 0.1306 | 0.1921 | 0.1258 | 20.3 | 4.6 | |
SYS | 0.0194 | 0.0264 | 0.0263 | 8.7 | 0.0757 | 0.1083 | 0.1073 | 6.3 | 5.0 | 0.0813 | 0.0275 | 0.0240 | 80.2 | 0.0870 | 0.1137 | 0.1124 | 8.0 | 26.0 | |
SCUSYS | 0.0194 | 0.0281 | 0.0188 | 29.6 | 0.0762 | 0.1090 | 0.1065 | 6.9 | 4.9 | 0.0290 | 0.0477 | 0.0227 | 52.0 | 0.0976 | 0.1365 | 0.1286 | 8.3 | 6.2 | |
DIF | 0.0410 | 0.0456 | 0.0455 | 11.5 | 0.1347 | 0.1591 | 0.1553 | 10.8 | 5.6 | 0.0666 | 0.0624 | 0.0617 | 17.6 | 0.2074 | 0.2137 | 0.2049 | 16.0 | 5.8 | |
SCUDIF | 0.0340 | 0.0531 | 0.0345 | 34.8 | 0.1140 | 0.1752 | 0.1065 | 21.8 | 4.5 | 0.0543 | 0.0864 | 0.0417 | 55.2 | 0.1656 | 0.2754 | 0.1102 | 39.3 | 4.0 | |
SYS | 0.0154 | 0.0201 | 0.0201 | 9.3 | 0.0447 | 0.0675 | 0.0672 | 5.1 | 5.6 | 0.0573 | 0.0196 | 0.0168 | 81.5 | 0.0642 | 0.0699 | 0.0693 | 14.4 | 22.4 | |
SCUSYS | 0.0149 | 0.0222 | 0.0145 | 31.2 | 0.0462 | 0.0695 | 0.0673 | 6.1 | 5.0 | 0.0237 | 0.0403 | 0.0130 | 59.7 | 0.1003 | 0.0963 | 0.0901 | 23.0 | 8.4 | |
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
DIF | 0.0447 | 0.0449 | 0.0454 | 15.8 | 0.1514 | 0.1007 | 0.1007 | 33.8 | 9.3 | 0.0637 | 0.0531 | 0.0534 | 22.5 | 0.1826 | 0.1102 | 0.1096 | 39.9 | 9.6 | |
SCUDIF | 0.0341 | 0.0499 | 0.0342 | 22.8 | 0.1078 | 0.1059 | 0.0848 | 28.3 | 7.3 | 0.0413 | 0.0615 | 0.0395 | 40.6 | 0.1150 | 0.1190 | 0.0850 | 31.8 | 6.9 | |
SYS | 0.0216 | 0.0313 | 0.0309 | 5.9 | 0.0937 | 0.0809 | 0.0804 | 21.6 | 6.7 | 0.0624 | 0.0496 | 0.0437 | 31.5 | 0.0810 | 0.1121 | 0.1025 | 10.7 | 10.9 | |
SCUSYS | 0.0226 | 0.0334 | 0.0222 | 23.1 | 0.0871 | 0.0817 | 0.0752 | 22.5 | 6.4 | 0.0266 | 0.0402 | 0.0273 | 35.6 | 0.0837 | 0.1072 | 0.0917 | 14.7 | 4.1 | |
DIF | 0.0244 | 0.0318 | 0.0318 | 8.3 | 0.0868 | 0.0879 | 0.0853 | 16.9 | 6.1 | 0.0313 | 0.0377 | 0.0374 | 10.7 | 0.1026 | 0.0978 | 0.0957 | 19.2 | 6.8 | |
SCUDIF | 0.0224 | 0.0338 | 0.0234 | 22.9 | 0.0741 | 0.0897 | 0.0728 | 18.8 | 5.4 | 0.0272 | 0.0409 | 0.0253 | 40.4 | 0.0790 | 0.1016 | 0.0739 | 22.7 | 6.0 | |
SYS | 0.0154 | 0.0222 | 0.0220 | 6.1 | 0.0405 | 0.0523 | 0.0507 | 9.1 | 5.0 | 0.0497 | 0.0319 | 0.0285 | 42.2 | 0.0782 | 0.0740 | 0.0687 | 25.2 | 14.2 | |
SCUSYS | 0.0161 | 0.0236 | 0.0161 | 23.8 | 0.0405 | 0.0523 | 0.0498 | 9.9 | 4.9 | 0.0184 | 0.0281 | 0.0185 | 37.6 | 0.0609 | 0.0840 | 0.0755 | 11.6 | 5.2 | |
DIF | 0.1627 | 0.0820 | 0.0797 | 54.4 | 0.3560 | 0.1615 | 0.1526 | 65.9 | 13.5 | 0.2329 | 0.0974 | 0.0941 | 70.2 | 0.4788 | 0.1926 | 0.1782 | 77.0 | 14.0 | |
SCUDIF | 0.0665 | 0.0994 | 0.0607 | 42.9 | 0.1477 | 0.1833 | 0.0848 | 44.2 | 6.9 | 0.0895 | 0.1302 | 0.0653 | 57.6 | 0.1725 | 0.2341 | 0.0851 | 50.8 | 6.0 | |
SYS | 0.0257 | 0.0370 | 0.0358 | 7.9 | 0.0936 | 0.0869 | 0.0857 | 18.9 | 6.3 | 0.0940 | 0.0287 | 0.0248 | 84.1 | 0.0591 | 0.0805 | 0.0792 | 7.2 | 17.5 | |
SCUSYS | 0.0281 | 0.0405 | 0.0249 | 38.8 | 0.0903 | 0.0891 | 0.0737 | 26.9 | 5.7 | 0.0452 | 0.0755 | 0.0305 | 61.7 | 0.0988 | 0.1274 | 0.0900 | 21.1 | 4.1 | |
DIF | 0.0432 | 0.0388 | 0.0383 | 19.4 | 0.1645 | 0.1288 | 0.1248 | 26.6 | 7.5 | 0.0642 | 0.0483 | 0.0468 | 27.5 | 0.2301 | 0.1623 | 0.1524 | 34.1 | 7.8 | |
SCUDIF | 0.0284 | 0.0437 | 0.0277 | 37.8 | 0.1012 | 0.1395 | 0.0720 | 34.7 | 5.4 | 0.0376 | 0.0589 | 0.0294 | 53.8 | 0.1259 | 0.1864 | 0.0730 | 44.6 | 5.3 | |
SYS | 0.0136 | 0.0188 | 0.0183 | 8.9 | 0.0348 | 0.0479 | 0.0471 | 7.2 | 5.2 | 0.0522 | 0.0190 | 0.0165 | 78.0 | 0.0552 | 0.0502 | 0.0499 | 21.7 | 16.9 | |
SCUSYS | 0.0140 | 0.0206 | 0.0133 | 33.9 | 0.0362 | 0.0492 | 0.0445 | 10.2 | 4.9 | 0.0195 | 0.0362 | 0.0135 | 54.7 | 0.0628 | 0.0881 | 0.0695 | 16.0 | 5.4 |
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
DIF | 0.0166 | 0.0231 | 0.0230 | 7.2 | 0.0752 | 0.0896 | 0.0879 | 12.0 | 4.5 | 0.0205 | 0.0265 | 0.0266 | 8.5 | 0.0813 | 0.0918 | 0.0900 | 12.8 | 4.8 | |
SCUDIF | 0.0171 | 0.0257 | 0.0175 | 21.6 | 0.0738 | 0.0894 | 0.0872 | 11.9 | 4.4 | 0.0197 | 0.0299 | 0.0195 | 37.0 | 0.0770 | 0.0918 | 0.0884 | 12.2 | 4.5 | |
SYS | 0.0125 | 0.0177 | 0.0175 | 5.9 | 0.0620 | 0.0822 | 0.0808 | 7.9 | 4.3 | 0.0327 | 0.0234 | 0.0210 | 35.8 | 0.0807 | 0.0988 | 0.0924 | 12.1 | 8.1 | |
SCUSYS | 0.0133 | 0.0193 | 0.0133 | 22.0 | 0.0616 | 0.0822 | 0.0801 | 8.2 | 4.3 | 0.0145 | 0.0209 | 0.0145 | 35.7 | 0.0733 | 0.0980 | 0.0936 | 8.9 | 3.9 | |
DIF | 0.0156 | 0.0219 | 0.0217 | 6.9 | 0.0530 | 0.0688 | 0.0684 | 8.7 | 5.2 | 0.0192 | 0.0251 | 0.0250 | 8.1 | 0.0590 | 0.0749 | 0.0739 | 10.2 | 5.3 | |
SCUDIF | 0.0167 | 0.0246 | 0.0173 | 25.2 | 0.0517 | 0.0688 | 0.0669 | 8.7 | 5.0 | 0.0197 | 0.0290 | 0.0174 | 44.1 | 0.0553 | 0.0760 | 0.0694 | 10.5 | 5.1 | |
SYS | 0.0119 | 0.0167 | 0.0164 | 6.5 | 0.0370 | 0.0545 | 0.0536 | 5.8 | 4.9 | 0.0430 | 0.0218 | 0.0195 | 57.7 | 0.1160 | 0.0683 | 0.0632 | 46.3 | 13.0 | |
SCUSYS | 0.0126 | 0.0184 | 0.0126 | 26.8 | 0.0368 | 0.0545 | 0.0527 | 6.1 | 5.0 | 0.0155 | 0.0214 | 0.0130 | 45.7 | 0.1134 | 0.0767 | 0.0705 | 38.0 | 7.5 | |
DIF | 0.0361 | 0.0335 | 0.0336 | 17.6 | 0.0953 | 0.0914 | 0.0921 | 16.5 | 6.1 | 0.0541 | 0.0410 | 0.0414 | 24.4 | 0.1149 | 0.0977 | 0.0978 | 21.0 | 6.3 | |
SCUDIF | 0.0267 | 0.0399 | 0.0260 | 37.9 | 0.0753 | 0.0933 | 0.0861 | 13.0 | 5.2 | 0.0347 | 0.0527 | 0.0270 | 53.8 | 0.0791 | 0.1022 | 0.0866 | 15.1 | 5.1 | |
SYS | 0.0136 | 0.0187 | 0.0187 | 7.7 | 0.0612 | 0.0795 | 0.0791 | 8.3 | 4.7 | 0.0760 | 0.0203 | 0.0180 | 91.7 | 0.0663 | 0.0825 | 0.0816 | 9.9 | 22.1 | |
SCUSYS | 0.0139 | 0.0204 | 0.0136 | 34.3 | 0.0610 | 0.0797 | 0.0786 | 8.4 | 4.6 | 0.0175 | 0.0271 | 0.0143 | 52.0 | 0.0701 | 0.0948 | 0.0908 | 8.9 | 4.6 | |
DIF | 0.0242 | 0.0257 | 0.0255 | 12.3 | 0.0718 | 0.0817 | 0.0789 | 12.8 | 5.2 | 0.0343 | 0.0317 | 0.0313 | 18.1 | 0.0958 | 0.0964 | 0.0911 | 16.7 | 5.2 | |
SCUDIF | 0.0202 | 0.0307 | 0.0192 | 39.2 | 0.0585 | 0.0865 | 0.0647 | 16.0 | 4.6 | 0.0270 | 0.0419 | 0.0197 | 56.6 | 0.0721 | 0.1109 | 0.0658 | 23.7 | 4.4 | |
SYS | 0.0111 | 0.0143 | 0.0142 | 9.8 | 0.0329 | 0.0489 | 0.0490 | 5.1 | 4.6 | 0.0545 | 0.0139 | 0.0123 | 94.8 | 0.0630 | 0.0476 | 0.0485 | 25.3 | 22.6 | |
SCUSYS | 0.0108 | 0.0162 | 0.0105 | 36.7 | 0.0332 | 0.0494 | 0.0490 | 5.5 | 4.5 | 0.0168 | 0.0258 | 0.0102 | 59.9 | 0.0874 | 0.0648 | 0.0629 | 29.5 | 7.0 | |
MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | MAE(θ) | SD(θ) | SE(θ) | Size(θ) | MAE(β) | SD(β) | SE(β) | Size(β) | RFSH | ||
DIF | 0.0319 | 0.0294 | 0.0291 | 18.2 | 0.1493 | 0.0638 | 0.0628 | 65.6 | 9.1 | 0.0393 | 0.0321 | 0.0319 | 23.2 | 0.1591 | 0.0660 | 0.0646 | 68.0 | 9.4 | |
SCUDIF | 0.0227 | 0.0346 | 0.0230 | 26.6 | 0.1247 | 0.0666 | 0.0577 | 56.9 | 8.0 | 0.0251 | 0.0386 | 0.0219 | 44.6 | 0.1253 | 0.0700 | 0.0579 | 56.7 | 7.7 | |
SYS | 0.0158 | 0.0227 | 0.0225 | 6.2 | 0.1107 | 0.0582 | 0.0571 | 49.0 | 6.2 | 0.0534 | 0.0340 | 0.0291 | 45.1 | 0.0498 | 0.0738 | 0.0674 | 7.9 | 9.0 | |
SCUSYS | 0.0176 | 0.0257 | 0.0173 | 27.5 | 0.1027 | 0.0592 | 0.0544 | 47.7 | 6.0 | 0.0190 | 0.0285 | 0.0180 | 43.2 | 0.0714 | 0.0696 | 0.0627 | 22.5 | 3.3 | |
DIF | 0.0148 | 0.0210 | 0.0206 | 6.7 | 0.0624 | 0.0472 | 0.0455 | 29.1 | 6.2 | 0.0168 | 0.0230 | 0.0226 | 8.0 | 0.0667 | 0.0495 | 0.0474 | 30.1 | 6.4 | |
SCUDIF | 0.0160 | 0.0238 | 0.0159 | 28.4 | 0.0579 | 0.0480 | 0.0434 | 29.1 | 6.1 | 0.0174 | 0.0264 | 0.0157 | 45.0 | 0.0595 | 0.0509 | 0.0439 | 30.3 | 5.8 | |
SYS | 0.0117 | 0.0165 | 0.0161 | 7.4 | 0.0372 | 0.0371 | 0.0360 | 17.7 | 5.2 | 0.0471 | 0.0226 | 0.0199 | 62.7 | 0.0476 | 0.0499 | 0.0454 | 18.4 | 10.8 | |
SCUSYS | 0.0130 | 0.0186 | 0.0125 | 31.2 | 0.0363 | 0.0370 | 0.0357 | 17.4 | 5.2 | 0.0141 | 0.0212 | 0.0131 | 43.9 | 0.0368 | 0.0529 | 0.0488 | 8.5 | 4.3 | |
DIF | 0.1016 | 0.0456 | 0.0444 | 63.9 | 0.2555 | 0.0851 | 0.0820 | 88.0 | 13.4 | 0.1297 | 0.0522 | 0.0502 | 74.1 | 0.2967 | 0.0954 | 0.0895 | 91.5 | 14.1 | |
SCUDIF | 0.0386 | 0.0580 | 0.0327 | 46.5 | 0.1287 | 0.0962 | 0.0576 | 57.2 | 8.3 | 0.0448 | 0.0696 | 0.0315 | 59.2 | 0.1331 | 0.1104 | 0.0577 | 57.6 | 8.3 | |
SYS | 0.0177 | 0.0256 | 0.0252 | 7.4 | 0.1147 | 0.0614 | 0.0610 | 46.7 | 7.0 | 0.0897 | 0.0209 | 0.0184 | 94.8 | 0.0384 | 0.0572 | 0.0564 | 5.4 | 18.4 | |
SCUSYS | 0.0210 | 0.0302 | 0.0170 | 45.2 | 0.1062 | 0.0638 | 0.0536 | 50.7 | 6.6 | 0.0254 | 0.0422 | 0.0174 | 58.9 | 0.0790 | 0.0754 | 0.0628 | 26.1 | 4.1 | |
DIF | 0.0231 | 0.0215 | 0.0210 | 18.1 | 0.0947 | 0.0631 | 0.0587 | 37.3 | 6.7 | 0.0294 | 0.0247 | 0.0238 | 23.2 | 0.1103 | 0.0710 | 0.0647 | 41.6 | 6.6 | |
SCUDIF | 0.0170 | 0.0255 | 0.0153 | 40.9 | 0.0619 | 0.0690 | 0.0431 | 35.4 | 5.7 | 0.0201 | 0.0303 | 0.0146 | 55.2 | 0.0673 | 0.0800 | 0.0434 | 39.0 | 5.5 | |
SYS | 0.0095 | 0.0131 | 0.0127 | 8.8 | 0.0321 | 0.0362 | 0.0347 | 13.5 | 4.8 | 0.0495 | 0.0135 | 0.0119 | 92.3 | 0.0470 | 0.0358 | 0.0353 | 28.9 | 17.1 | |
SCUSYS | 0.0105 | 0.0154 | 0.0094 | 40.2 | 0.0329 | 0.0369 | 0.0331 | 17.3 | 4.9 | 0.0122 | 0.0208 | 0.0092 | 54.6 | 0.0383 | 0.0550 | 0.0482 | 10.9 | 4.3 |
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Ashley, R.A.; Sun, X. Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models. Econometrics 2016, 4, 47. https://doi.org/10.3390/econometrics4040047
Ashley RA, Sun X. Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models. Econometrics. 2016; 4(4):47. https://doi.org/10.3390/econometrics4040047
Chicago/Turabian StyleAshley, Richard A., and Xiaojin Sun. 2016. "Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models" Econometrics 4, no. 4: 47. https://doi.org/10.3390/econometrics4040047
APA StyleAshley, R. A., & Sun, X. (2016). Subset-Continuous-Updating GMM Estimators for Dynamic Panel Data Models. Econometrics, 4(4), 47. https://doi.org/10.3390/econometrics4040047