Rescaling Residual Bootstrap and Wild Bootstrap
International Journal of Data Science and Analysis
Volume 2, Issue 1, October 2016, Pages: 7-14
Received: Jul. 20, 2016; Accepted: Oct. 14, 2016; Published: Oct. 28, 2016
Views 3771      Downloads 137
Acha Chigozie Kelechi, Department of Statistics, Michael Okpara University of Agriculture, Umudike, Nigeria
Article Tools
Follow on us
This paper examines and discusses a comparative analysis of hypothetical data by using bootstrap methods. The residual and wild bootstrap methods, including their rescaled versions were applied on the data collected from a normal distribution with different ability levels to check whether they are significant at various assessment conditions. The wild bootstrap compared in this paper are from Mammen and Redamarche distributions. In addition their kernel density plot is used to ascertain the trends and the performance at the lower ends of the distributions for each bootstrap model and also the trend as sample size tends to infinity. To achieve this, each of the forms were represented by using at least one functional model each from hypothetical data sets of a particular bootstrap data generating process (DGP) method to illustrate how 8640 scenerios were estimated. The result shows that the Hypothetical Rescaled Residual (HRR) is found to be preferable to the Hypothetical Unrescaled Residual (HR) while Hypothetical Wild Redamarche Model (HRWR) is found to be preferable to the Hypothetical Wild Mammen model (HRWM) with reference to their bias, standard error and root mean square error (RMSE) at different levels of significance, that is, B=99, N(0,1), n1 & n3 = 10000, RMSE = -0.0004 &-0.0025 respectively. Aslo, B=99, N(0,1), n3 = 10000, RMSE = -0.0004. Even though at B=99, N(0,1), n2 = 10000, RMSE for HRWM (0.0601) is higher than HRWR (0.0595). In fact, across all the models, rescaled residual functional model out performed all other functional models considered in this paper. Also, the trends at the lower ends of the distributions for each bootstrap model shows that the empirical distributions of true distributions follow the chi-square distribution and also tends to normal distribution as sample size tends to inifinity.
Rescaled, Bootstrap, Hypothetical Models, Mammen Distributions, Redamarche Distributions
To cite this article
Acha Chigozie Kelechi, Rescaling Residual Bootstrap and Wild Bootstrap, International Journal of Data Science and Analysis. Vol. 2, No. 1, 2016, pp. 7-14. doi: 10.11648/j.ijdsa.20160201.12
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Acha, C. K. (2014) Parametric Bootstrap Methods for Parameter Estimation in SLR Models. International Journal of Econometrics and Financial Management, 2 (5), 175–179. Doi: 10.12691/ijefm-2-5-2.
Acha, C. K. (2014) Bootstrapping Normal and Binomial Distributions. International Journal of Econometrics and Financial Management, 2 (6), 253– 256. Doi: 10.12691/ijefm-2-6-2.
Acha, C. K. and Acha I. A. (2015) Smooth Bootstrap Methods on External Sector Statistics. International Journal of Econometrics and Financial Management, 3 (3), 115–120. Doi: 10.12691/ijefm-3-3-2.
MacKinnon, J. G., (2006). Bootstrap Methods in Econometrics, The Economic Record, The Economic Society of Australia, 82 (1), 2-18.
Davidson, R. and Flachaire, E. (2001), The Wild Bootstrap, Tamed at Last, GREQAM Document de Travail 99A32, revised.
Davidson, R. and Flachaire, E., (2008). The wild bootstrap, tamed at last, Journal of Econometrics, Elsevier, 146 (1), 162-169.
Lahiri, S. N. (2006). Bootstrap Methods: A Review. In Frontiers in Statistics (J. Fan and H. L. Koul, editors) 231-265, Imperial College Press, London.
Flachaire, E. (2005). More efficient tests robust to heteroskedasticity of unknown form. Econ. Rev. 24, 219–241.
Godfrey, L. G. (1998). Tests of non-nested regression models: Some results on small sample behaviour and the bootstrap, Journal of Econometrics, 84, 59–74.
Godfrey, L. G., and Veall, M. R. (2000). Alternative approaches to testing by variable addition. Econ. Rev. 19, 241–261.
Davidson, R. and MacKinnon, J. G. (1996). The Power of Bootstrap Tests, Working Papers 937, Queen's University, Department of Economics.
Davidson, R & MacKinnon, J. G. (1999). The Size Distortion of Bootstrap Tests, Econometric Theory, Cambridge University Press, vol. 15 (03), pages 361-376, June.
Davidson, R. and MacKinnon, J. (2000). Bootstrap tests: how many bootstraps?, Econometric Reviews, Taylor and Francis Journals, 19 (1), 55-68.
Davidson, R. and MacKinnon, J. G., (2006a). The power of bootstrap and asymptotic tests, Journal of Econometrics, Elsevier, 133 (2), 421-441.
Davidson, R. and MacKinnon, J. G. (2006b), ‘Bootstrap Methods in Econometrics’, in Patterson, K. and Mills, T. C. (eds), Palgrave Handbook of Econometrics: Volume 1 Theoretical Econometrics. Palgrave Macmillan, Basingstoke; 812–38.
Acha, I. A. and Acha, C. K. (2011). Interest Rates in Nigeria: An Analytical Perspective. Research Journal of Finance and Accounting, 2 (3); 71-81 ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online).
Acha, C. K. and Omekara, C. O. (2016) Towards Efficiency in the Residual and Parametric Bootstrap Techniques. American Journal of Theoretical and Applied Statistics. 5 (5) 285-289. doi: 10.11648/j.ajtas.20160505.16.
Albert, J., and Chib, S. (1993) ‘Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts”, Journal of Business and Economic Statistics 11, 1–15.
Beran, R. (1988), ‘Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements’, Journal of the American Statistical Association, 83, 687–97.
Hansen, B. E. (2000). Testing for structural change in conditional models. J. Econ. 97, 93–115.
Geweke, J. (1999) ‘Using simulation methods for Bayesian econometric models: Inference, development and communication’ (with discussion and reply), Econometric Reviews 18, 1–126.
Wu, C. F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Annals of. Statistics. 14, 1261-1295.
McCulloch, R. and P. Rossi, P. (1994) ‘An exact likelihood analysis of the multinomial probit model,’ Journal of Econometrics 64, 207–40.
MacKinnon, J. G. and Smith, A. A. (1998). Approximate bias correction in econometrics, Journal of Econometrics, Elsevier, 85 (2), 205-230.
MacKinnon J. G. (2002), Bootstrap inference in econometrics, Canadian Journal of Economics Revue Canadienne dEconomique, 35 (4): 615-645. On-line: 10.1111/0008-4085.00147.
Mammen, E. (1992). Bootstrap and wild bootstrap for high dimensional linear models. Ann. Statist. 21, 255–285.
Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. Ann. Statist. 21, 255–285.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186