  In this paper we propose a variable bandwidth kernel regression estimator for
$i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical
Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$
under the condition that the fifth order derivative of the density function and
the sixth order derivative of the regression function are bounded and
continuous. We also establish the central limit theorems for the proposed ideal
and true variable kernel regression estimators. The simulation study confirms
our results and demonstrates the advantage of the variable bandwidth kernel
method over the classical kernel method.
