A Class of Improved Parametrically Guided Nonparametric Regression Estimators

Abstract

In this article we define a class of estimators for a nonparametric regression model with the aim of reducing bias. The estimators in the class are obtained via a simple two-stage procedure. In the first stage, a potentially misspecified parametric model is estimated and in the second stage the parametric estimate is used to guide the derivation of a final semiparametric estimator. Mathematically, the proposed estimators can be thought as the minimization of a suitably defined Cressie–Read discrepancy that can be shown to produce conventional nonparametric estimators, such as the local polynomial estimator, as well as existing two-stage multiplicative estimators, such as that proposed by Glad (1998). We show that under fairly mild conditions the estimators in the proposed class are asymptotically normal and explore their finite sample (simulation) behavior.

Keywords:

• Asymptotic normality
• Combined semiparametric estimation

JEL Classification:

• C14
• C22

ACKNOWLEDGMENTS

We thank participants of the Second Conference on Information and Entropy Econometrics, Amos Golan, Essie Maasoumi, Peter Phillips, Jeff Racine, and an anonymous referee for helpful comments. The authors retain responsibility for any remaining errors.

Notes

¹Glad (1998) established the order of the bias and variance for her estimator, but no result on its asymptotic distribution.

²See Cressie and Read (1984) and Read and Cressie (1988).

³If φ is such that E(φ(Z _i (x), b ₀)|x _i ) = 0 for a unique b ₀, with φ = (Z _i (x) − b ₀)K _{h n} (x _i  − x), then the maximization in (9) gives and provided that .

⁴Mutatis mutandis local polynomial estimators of order p ≥ 2 can also be obtained from the optimization in (12).

⁵Since the parametric guide is linear, the LL and additively corrected estimators coincide.

⁶Here θ₀ is calculated by minimizing the Kullback–Leibler discrepancy or maximizing the likelihood function. For α = 1, 0 the bias term doesn't involve θ₀.

⁷The choice of α is guided by the intention of including both positive and negative values of α, as well as taking into account the special cases where α = 0 and α = 1 that correspond to the additively corrected estimator and that of Glad (1998).

⁸Suitable estimators are given in the comments following Theorem 2.

Note: All entries for bias squared, variance, and mean square error are multiplied by 10⁴.

Note: All entries for bias squared, variance and mean square error are multiplied by 10⁴.

⁹As we do not prove the strict convexity of MISE with respect to α there may be several α that minimize MISE.

¹⁰Under the unrealistic assumption of a correctly specified parametric DGP, a suitable parametric estimator (possibly unbiased and efficient in an appropriately defined class) can be chosen, and the bias-variance tradeoff intrinsic to all nonparametric estimators considered herein can be bypassed.

¹¹When the parametric guide is equal to m(x) the left-hand side of Eq. (15) is identically zero for all x.

¹²In fact, optimal values of α do not vary significantly with n for n ≥ 100 in our simulations.