Journal of Statistical Planning and Inference 137 (2007) 379 – 404 www.elsevier.com/locate/jspi Nonparametric regression with heteroscedastic long memory errors Hongwen Guo, Hira L. Koul ∗ Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824-1027, USA Received 24 May 2005; received in revised form 30 January 2006; accepted 30 January 2006 Available online 4 May 2006 Abstract This paper discusses the estimation of regression and variance functions in nonparametric heteroscedastic regression models with long memory moving average errors and uniform nonrandom design on the unit interval. The consistency and the finite dimensional weak convergence of these estimators are established. For the regression function estimators, the asymptotic normality is established for all values of the long memory parameter 1 2 <H< 1; while for the variance function estimators, the asymptotic normality is proved for 1 2 <H< 3 4 , nonnormality for 3 4 <H< 1. The paper also establishes the uniform convergence rate of the regression function estimators to be (nb) 1-H / log n for all 1 2 <H< 1 and for a large class of innovations, including bounded and Gaussian innovations, where n is the series size and b is the bandwidth used in estimating the regression function. Additionally, the local Whittle estimator of H based on standardized nonparametric residuals is shown to be log(n)-consistent and the finite dimensional distributions of the studentized versions of the regression function estimators are shown to be asymptotically normal. These results thus generalize some of the results of Robinson to heteroscedastic regression models with long memory moving average errors. © 2006 Elsevier B.V.All rights reserved. MSC: Primary 62G07; secondary 62M10 Keywords: Moving average errors; Local Whittle; Nonparametric residuals 1. Introduction A stochastic process is said to have long memory if its lag k auto-covariances decay to zero like k - , for some 0 <  < 1. Long memory processes have been found to arise in a variety of physical and social sciences, see, e.g. Beran (1994), Baillie (1996), Dehling et al. (2002), Doukhan et al. (2003), Robinson (2003), and references therein. On the other hand, nonparametric heteroscedastic regression models are also found to be very useful in practice. More precisely, consider the model Y t = r  t n  +   t n  u t , t = 1, 2,...,n, (1.1) ∗ Corresponding author. E-mail address: koul@stt.msu.edu (H.L. Koul). 0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.01.016