Volume 19, Issue 2 (12-2020)                   JIRSS 2020, 19(2): 15-31 | Back to browse issues page

XML Print

Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran. , sheikhy.a@uk.ac.ir
Abstract:   (78 Views)

In this paper we obtain a consistent estimator when there exist some measurement errors and multicollinearity in the instrumental variables in a two stage least square estimation of parameters. We investigate the asymptotic distribution of the proposed estimator and discuss its properties using some theoretical proofs and a simulation study. A real numerical application is also provided for more illustration.

Full-Text [PDF 164 kb]   (8 Downloads)    
Type of Study: Original Paper | Subject: 62Jxx: Linear inference, regression
Received: 2019/08/17 | Accepted: 2021/04/17 | Published: 2020/12/11

1. Akdeniz, F., and Roozbeh, M. (2017), Efficiency of the generalized-difference-based weighted mixed almost unbiased two-parameter estimator in partially linear model. Communications in Statistics - Theory and Methods, 46(24), 12259-12280. [DOI:10.1080/03610926.2017.1295075]
2. Arashi, M.,Tabatabaey, S. M. M., and Hassanzadeh Bashtian, M. (2014), Shrinkage ridge estimators in linear regression. Communication in Statistics-Simulation and Computation, 43, 871-904. [DOI:10.1080/03610918.2012.718838]
3. Bowden, R., and Turkington, D. (1984), Instrumental variables. Cambridge University Press, NY. [DOI:10.1017/CCOL0521262410]
4. Burgess, S., Small, D. S., and Thompson, S. G. (2017), A review of instrumental variable estimators for Mendelian randomization. Statistical Methods in Medical Research, 26(5), 2333-2355. [DOI:10.1177/0962280215597579]
5. Burgess, S. and Dylan, S. (2016), Predicting the Direction of Causal Effect Based on an Instrumental Variable Analysis: A Cautionary Tale. J. Causal Infer., 4(1), 49-59. [DOI:10.1515/jci-2015-0024]
6. Cao, C. Z., Lin, J. G., Shi, J. Q., Wang, W., and Zhang, X. Y. (2015), Multivariate measurement error models for replicated data under heavy-tailed distributions. Chemometrics, 8, 457-466. [DOI:10.1002/cem.2725]
7. Carrasco, M. (2012). A regularization approach to the many instruments problem. Journal of Econometrics, 170(2), 383-398. [DOI:10.1016/j.jeconom.2012.05.012]
8. Carrasco, M., and Tchuente, G. (2015), Regularized LIML for many instruments. Journal of Econometrics, 186(2), 427-442. [DOI:10.1016/j.jeconom.2015.02.018]
9. Carroll, R., Ruppert, J. D., Stefanski, L. A., and Crainiceanu, C. M. (2006), Measurement Error in Nonlinear Models: A Modern Perspective, CRC Press, Boca Raton, Fla, 2nd edition, pp. 485. [DOI:10.1201/9781420010138]
10. Chan, L. K., and Mak, T. K. (1979), On the maximum likelihood estimation of a linear structural relationship when the intercept is known. J. Multivar. Anal., 9, 304-313. [DOI:10.1016/0047-259X(79)90087-3]
11. Devanarayana, V., and Stefanski, L. A. (2002), Empirical Simulation Extrapolation for Measurement Error Models With Replicate Measurements, Statistics and Probability Letters, 59, 219-225. [DOI:10.1016/S0167-7152(02)00098-6]
12. Didelez, V., and Sheehan, N. A. (2007), Mendelian randomization as an instrumental variable approach to causal inference. Stat. Meth. Med. Res., 16, 309-330. [DOI:10.1177/0962280206077743]
13. Ebbes, P. (2004), Latent Instrumental Variables - A New Approach to Solve for Endogeneity. Ph.D.thesis, University of Groningen Economics and Business.
14. Golub, G. H., Heath, M., and Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21, 215-223. [DOI:10.1080/00401706.1979.10489751]
15. Lawlor, D. A., Harbord, R. M., Sterne, J. A., Timpson, N., and Smith, G. D. (2008), Mendelian randomization: using genes as instruments for making causal inferences in epidemiology. Statistics in Medicine, 27(8), 1133-1163. [DOI:10.1002/sim.3034]
16. Liu, X. Q., and Jiang, H. Y. (2012), Optimal generalized ridge estimator under the generalized cross-validation criterion in linear regression. Linear Algebra and its Applications, 436(5), 1238-1245. [DOI:10.1016/j.laa.2011.08.032]
17. Fallah, R., Arashi, M., and Tabatabaey, S. M. M. (2017), On the ridge regression estimator with sub- space restriction, Communication in Statistics - Theory and Methods, 46(23), 11854-11865. [DOI:10.1080/03610926.2017.1285928]
18. Ghapani, F., Babadi, B. (2016), A New Ridge Estimator in Linear Measurement Error Model with Stochastic Linear Restrictions. JIRSS, 15(2), 87-103. [DOI:10.18869/acadpub.jirss.15.2.87]
19. Hausman, J. A. (1978), Specification tests in econometrics. Econometrica, 46, 1251-1271. [DOI:10.2307/1913827]
20. Hansen, C., and Kozbur, D. (2014), Instrumental Variables Estimation with Many Weak Instruments Using Regularized JIVE. Journal of Econometrics, 182(2), 290-308. [DOI:10.1016/j.jeconom.2014.04.022]
21. Harris, M. H., Gold, D. R., Rifas-Shiman, S. L., Melly, S.J., Zanobetti, A., Coul, B. A., Schwartz, J. D., Gryparis ,A., Kloog, I., and Koutrakis, P. (2016), Prenatal and childhood traffic-related air pollution exposure and childhood executive function and behavior. Neurotoxicol. Teratol., 57, 60-70. [DOI:10.1016/j.ntt.2016.06.008]
22. Horel, A. E., and Kennard, R. W. (1970), Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55- 83. [DOI:10.1080/00401706.1970.10488634]
23. Isogawa, Y. (1985), Estimating a multivariate linear structural relationship with replication. J. R. Statist. Soc. B, 47, 211-215. [DOI:10.1111/j.2517-6161.1985.tb01347.x]
24. Lachos, B. G., and Bolfarine, H. (2016), Heavy tailed calibration model with Berkson measurement errors for replicated data. Chemometrics and Intelligent Laboratory systems, 156, 21-35. [DOI:10.1016/j.chemolab.2016.04.014]
25. Lousdal, M. L. (2018), An introduction to instrumental variable assumptions, validation and estimation. Emerging themes in epidemiology, 15(1), 1-7. [DOI:10.1186/s12982-018-0069-7]
26. Nawarathna, L. S., and Choudhary, P. K. (2015), A Heteroscedastic Measurement Error Model for Method Comparison Data With Replicate Measurements. Statistics in Medicine, 34(7), 1242-1258. [DOI:10.1002/sim.6424]
27. Okbay, A. et al. (2016), Genome-wide association study identifies 74 loci associated with educational attainment. Nature.
28. Roozbeh, M., Hesamian, G., Akbari, M. G., (2020), Ridge estimation in semi-parametric regression models under the stochastic restriction and correlated elliptically contoured errors. J Comput. Appl. Math., 378, 112940. [DOI:10.1016/j.cam.2020.112940]
29. Rasekh, A. R. (2006), Local influence in measurement error models with ridge estimate. comput. Stat. Data Anal, 50, 2822-2834. [DOI:10.1016/j.csda.2005.04.022]
30. Rasekh, A. R., Fieller, N. R. J. (2003), Influence functions in functional measurement error models with replicated data. Statistics, 37(2), 169-178. [DOI:10.1080/0233188031000112890]
31. Rietveld, C. A. et al. (2013), GWAS of 126,559 Individuals identifies genetic variants associated with educational attainment. Science, 340(6139), 1467-1471.
32. Saleh, A. M. E., Arashi, M., and Kibria, G. (2019), Theory of Ridge Regression Estimation with Applications. John Wiley, USA. [DOI:10.1002/9781118644478]
33. Saleh, E., and Shalabh (2014), Ridge regression estimation approach to measurement error model, J. Multivariate Analysis., 123, 68-84. [DOI:10.1016/j.jmva.2013.08.014]
34. Sheikhi, A., Bahador, F., Arashi, M. (2020), On a generalization of the test of endogeneity in a two stage least squares estimation. Journal of Applied Statistics, DOI: 10.1080/02664763.2020.1837084. [DOI:10.1080/02664763.2020.1837084]
35. Singh, S., Jain, K., Sharma, S. (2014), Replicated measurement error model under exact linear restrictions. Statist Pap., 55, 253-274. [DOI:10.1007/s00362-012-0469-7]
36. Martens, E. P., Pestman, W. R., de Boer, A., Belitser, S. V., and Klungel, O. H. (2006), Instrumental variables: application and limitations. Epidemiology, 17, 260-267. [DOI:10.1097/01.ede.0000215160.88317.cb]
37. Shalabh, C. M., and Paudel, N. K. (2009), Consistent estimation of regression parameters under replicated ultrastructural model with non normal errors. J. Stat Comp Sim, 79(3), 251-274. [DOI:10.1080/00949650701748556]
38. Wooldridge, J. M. (2016), Introductory Econometrics: A Modern Approach 6th Edition. Cengage Learning, Boston, MA.
39. Wu, D. M. (1973), Alternative tests of independence between stochastic regressors and disturbances. Econometrica, 42, 529-546. [DOI:10.2307/1911789]