Ioannides (1992) investigated the asymptotic properties of the integrated square error (ISE) of general kernel estimators of the unknown regression function in nonparametric regression with independent random errors. It is well known, however, that the assumption of independent errors is often violated in practical situations, especially in the analyses of economic data. In this article, we relax this assumption by modeling the errors with a moving average process of infinite order, and establish the asymptotic normality and strong consistency of the ISE by extending the martingale central limit theorem. These results can be used to construct test statistics and make asymptotically efficient statistical inference in nonparametric regressions with serially correlated errors.
- Central limit theorem
- Integrated square error (ISE)
- Kernel smoothing
- Law of large numbers
- Nonparametric estimators
- Nonparametric regression function
- Serially correlated errors