Journal of International

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1 ISSN Journal of International Academic Research for Multidisciplinary A Scholarly, Peer Reviewed, Monthly, Open Access, Online Research Journal Impact Factor VOLUME 1 ISSUE 11 DECEMBER 2013 A GLOBAL SOCIETY FOR MULTIDISCIPLINARY RESEARCH A GREEN PUBLISHING HOUSE

2 Editorial Board Dr. Kari Jabbour, Ph.D Curriculum Developer, American College of Technology, Missouri, USA. Er.Chandramohan, M.S System Specialist - OGP ABB Australia Pvt. Ltd., Australia. Dr. S.K. Singh Chief Scientist Advanced Materials Technology Department Institute of Minerals & Materials Technology Bhubaneswar, India Dr. Jake M. Laguador Director, Research and Statistics Center, Lyceum of the Philippines University, Philippines. Prof. Dr. Sharath Babu, LLM Ph.D Dean. Faculty of Law, Karnatak University Dharwad, Karnataka, India Dr.S.M Kadri, MBBS, MPH/ICHD, FFP Fellow, Public Health Foundation of India Epidemiologist Division of Epidemiology and Public Health, Kashmir, India Dr.Bhumika Talwar, BDS Research Officer State Institute of Health & Family Welfare Jaipur, India Dr. Tej Pratap Mall Ph.D Head, Postgraduate Department of Botany, Kisan P.G. College, Bahraich, India. Dr. Arup Kanti Konar, Ph.D Associate Professor of Economics Achhruram, Memorial College, SKB University, Jhalda,Purulia, West Bengal. India Dr. S.Raja Ph.D Research Associate, Madras Research Center of CMFR, Indian Council of Agricultural Research, Chennai, India Dr. Vijay Pithadia, Ph.D, Director - Sri Aurobindo Institute of Management Rajkot, India. Er. R. Bhuvanewari Devi M. Tech, MCIHT Highway Engineer, Infrastructure, Ramboll, Abu Dhabi, UAE Sanda Maican, Ph.D. Senior Researcher, Department of Ecology, Taxonomy and Nature Conservation Institute of Biology of the Romanian Academy, Bucharest, Romania Dr. Reynalda B. Garcia Professor, Graduate School & College of Education, Arts and Sciences Lyceum of the Philippines University Philippines Dr.Damarla Bala Venkata Ramana Senior Scientist Central Research Institute for Dryland Agriculture (CRIDA) Hyderabad, A.P, India PROF. Dr.S.V.Kshirsagar, M.B.B.S,M.S Head - Department of Anatomy, Bidar Institute of Medical Sciences, Karnataka, India. Dr Asifa Nazir, M.B.B.S, MD, Assistant Professor, Dept of Microbiology Government Medical College, Srinagar, India. Dr.AmitaPuri, Ph.D Officiating Principal Army Inst. Of Education New Delhi, India Dr. Shobana Nelasco Ph.D Associate Professor, Fellow of Indian Council of Social Science Research (On Deputation}, Department of Economics, Bharathidasan University, Trichirappalli. India M. Suresh Kumar, PHD Assistant Manager, Godrej Security Solution, India. Dr.T.Chandrasekarayya,Ph.D Assistant Professor, Dept Of Population Studies & Social Work, S.V.University, Tirupati, India.

3 ESTIMATION OF AR (1) COEFFICIENT IN THE CONTEXT OF ERRORS IN VARIABLES DR. PROF. V. BALAKRISHNAMA NAIDU* DR. PROF. V. KODANDARAMI REDDY** *Professor & Head, Dept. of Econometrics, S.V. University, Tirupati, India **Professor, Dept. of Econometrics, S. V. University, Tirupati, A.P, India ABSTRACT In a linear regression model, when errors are auto-correlated and further if there are errors in variables, no estimator of first order autocorrelation coefficient was provided in the econometric literature. This paper studies just a comparison of three different estimators of serial correlation in the presence of measuremental errors using a simple Monte Carlo experiment. None of the estimators turns out to be superior to the others over the entire range of parameter ρ. But there is some gain in efficiency to be had from using the Durbin procedure in the presence of high levels of serial correlation in the residuals and little loss from using such methods when the true ρ is small. KEY WORDS: Autocorrelation, Measuremental Errors, Monte Carlo Experiment, Mean Square Error (MSE) INTRODUCTION It is well know that errors in variables cause least squares estimators of regression coefficients to be biased and inconsistent, but there does not seem to be any treatment in the econometric literature to the problem of errors in variables in the context of auto correlated disturbances. The effect of errors in variables on the estimated first order autocorrelation was studied by Lankipalle & et al (1985) and observed that the conventional estimator of ρ using the least squares residuals under estimate the true parameter ρ. Thus in the presence of measuremental errors, it is not possible to make correct assessment of serial correlation of the residuals using the OLS estimator of ρ. Various methods of estimating ρ, viz., least squares estimate, Cochrane-Orcutt and Durbin etc., are proposed when there are no measuremental errors. Since no method was proposed for estimating ρ in the presence of measuremental errors, the above three methods are considered even in the presence of measuremental errors and in this paper an attempt has been made to assess the nature of bias in the estimate of ρ under the three methods using a simple Monte Carlo experiment. MONTE CARLO RESULTS: A Monte Carlo experiment is conducted to assess the nature of bias in the estimator of the first order autocorrelation coefficient in the presence of measuremental errors. The model used for the sampling experiment is 667

4 Y t = X t + W t (1) where W t = ρ W t-1 + Є t (2) and X * t = X t + U t (3) Y * t = Y t + V t (4) * where X t and Y * t are the observed values with the measuremental errors U t and V t respectively. For a given ρ, W t is generated using equation (2). To ensure that the error series was stationary, W 1 is generated as W 1 =. Є 1 1.0/ (1- ρ 2 ) The variables X, U, V and Є are generated with the following specifications using the Box-Muller (1958) method. X t ~ N (0, 0.025), U t ~ N (0, 0.05), V t ~ N (0, 0.05) and Є t ~ N (0, 0.06) The observations on X * t and Y * t are generated using the equations (3) and (4). Samples of sizes 10, 20 and 50 are generated and eight different values of ρ are considered. To ensure accurate results each experiment involved 50 replications. COMPARISON OF VARIOUS ESTIMATORS OF ρ: The Monte Carlo information on the bias of the estimator of ρ in the presence of measuremental errors for the three methods under consideration is summarized in Table-1. Table-2 summarizes the results in the case of no measuremental errors. It is observed that the bias in the estimate in all the methods is increasing as the value of true ρ increases, on either direction. But increase is more on the positive direction of ρ rather than on the negative direction of ρ. Further it is noticed that the bias in the estimate decreases as the sample size increases except in Cochrane-Orcutt method where the bias is almost the same. In general it is observed that in all sample sizes and for different true values of ρ the bias in the estimate is comparatively low in the Durbin s method. Next comes to the method of conventional estimator of ρ. For small values of ρ the bias in the estimator using the OLS and Durbin procedures is somewhat low. The superiority of the Durbin estimator over the other two estimators in the entire range of ρ is only slight. Further the bias in the estimate of ρ is low when autocorrelation is negative when compared to the bias if the autocorrelation is positive. Under the Durbin s method of estimating ρ, if the true ρ is around -0.2, the ratio of bias to is 28 per cent in the case of sample size 10, 15 per cent in the case of sample size 20, and 25 per cent in the case of sample size 50. Thus we notice that for moderate sample of size 20 to 25 the ratio is around 15 per cent whereas for other methods the ratio is more. 668

5 When there are no measuremental errors, it is observed that the bias decreases as the sample size increases. Further the bias in Durbin s estimate of ρ is less when compared with other two methods. These results confirm with the results of Rao and Griliches (1969) when there are no measuremental errors. The performance of a particular ρ depends, however, not only on its average bias but also on its variance. Generally a less biased estimator may have a higher variance cancelling much of the gain from the reduction in bias. To investigate this, we computed the mean square errors (MSE) for each ρ over the 50 samples. In order to convert them into comparable units, we divided the MSE s by the MSE s of other ρ s. These ratios are presented in the tables 7 and 8. From the tables we observe that Durbin s ρˆ is better for high values of ρ (either +ve or ve), while at the same time not being distinctly inferior to the other two methods for small values of ρ in large samples. Even in the case of no measuremental errors, Durbin ρˆ is better for high values of ρ. This observation confirms with the observation made by Rao and Griliches (1969). When compared to the bias with measuremental errors and without measuremental errors it is clear that there is a reduction in bias and when there are no measuremental errors. This clearly shows the effect of errors in variables on the estimator of ρ. CONCLUSION Among the various estimators examined in the presence of measuremental errors, we can notice that Durbin s estimator of ρ is likely to be the best over a wide range of true parameter ρ, than other methods. Also there is definite decrease in bias, MSE, and ratio of bias to by using Durbin s estimator for negative values of ρ. Further, it is observed from the ratios of MSE s that Durbin s ρˆ is better for higher values of ρ (either +ve or ve), while at the same time not being distinctly inferior to the other two methods for small values of ρ in large samples. Hence in the absence of a consistent estimator of ρ in the presence of the measurement errors, one may use Durbin s procedure of estimating first order serial correlation among the residuals. REFERENCES 1. G.E.P. Box and M.E. Muller, A Note on the Generation of Random Normal Deviates, Annals of Mathematical Statistics, Vol. 29, 1958, pp Lankipalle, K.N, Reddy, V.K, and Naidu, V.B, Errors in Variables and Autocorrelation, Proceedings of the Regional Conference of the Indian Econometric Society, Tirupati, October Potluri Rao and Zvi Griliches, Small Sample Properties of Several Two Stage Regression Methods in the Context of Auto correlated Errors, Journal of the American Statistical Association, Vol. 64, 1969, pp

6 Table - 1 : Bias and of Different Estimators of RHO with Measuremental Errors TRUE RHO METHOD Sample Size 10 Sample Size 20 Sample Size 50 TO TO TO L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin

7 Table - 2 : Bias and of Different Estimators of RHO without Measuremental Errors TRUE RHO METHOD Sample Size 10 Sample Size 20 Sample Size 50 TO TO TO L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin L.S.E Cochrane Durbin

8 Table -3 Ratio's of Relative Mean Square Errors of Different Estimators of RHO True RHO COCHRAN Sample Size 10 Sample Size 20 Sample Size 50 COCHRAN/ COCHRAN With Measuremental Errors COCHRAN/ COCHRAN COCHRAN/ Without Measuremental Errors

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