In this paper we focus on nonparametric analysis of the volatility function for mixing processes. Our approach is based on local polynomial smoothing and supplies several tools which can be used to test a specific parametric model: nonparametric function estimation, nonparametric confidence intervals, and nonparametric test for symmetry. At the same time, it faces the main drawbacks of the nonparametric procedures proposed so far in the literature that are the choice of the bandwidth parameter and the slow convergence rate. Another aim of this paper is to focus on the advantages of an alternative representation for the parametric GARCH(1,1) model in terms of a Nonparametric-ARCH(1) model, to be estimated by local polynomials. We prove the consistency of the proposed method and investigate its empirical performance on synthetic and real datasets.
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