برآورد غیر پارامتریک و نیمه پارامتریک و آزمون الگوهای اقتصادسنجی با پارامترهای هموارسازی وابسته به داده ها

We consider nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Most of the existing works on asymptotic distributions of a nonparametric/semiparametric estimator or a test statistic are based on some deterministic smoothing parameters, while in practice it is important to use data-driven methods to select the smoothing parameters. In this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters. We also allow for general weakly dependent data.

There is a rich literature on using nonparametric techniques to estimate and test for statistical/econometric models. It is well known that the selection of smoothing parameters is of crucial importance in nonparametric estimation and testing. Various data-driven methods have been proposed in the literature. The least squares cross-validation method is one of the most popular approach used by applied researchers. For density estimation with the kernel method, Stone (1984) has established the optimality result for the least squares cross-validation method under quite weak regularity conditions. Li (1987) provides general optimality results for various data-driven methods with nonparametric series and kk-nearest neighbor estimation methods. The optimality results of the cross-validation method in the regression model framework have been studied by Härdle and Marron (1985) and Härdle et al., 1988 and Härdle et al., 1992. Recently, Hall et al., 2004 and Hall et al., 2007 have shown that the least squares cross-validation method has the additional advantage of being able to remove irrelevant covariates in nonparametric kernel conditional density and regression function estimations. However, the literature of nonparametric estimation and testing mainly focus on using deterministic (non-stochastic) smoothing parameters when working with the asymptotic distribution of a nonparametric/semiparametric estimator or of a test statistic. Ichimura (2000) considers the asymptotic distribution of nonparametric/semiparametric estimators with data dependent smoothing parameters. Ichimura uses results developed in Pollard, 1984 and Pollard, 1990 and Sherman, 1994a and Sherman, 1994b to establish stochastic equicontinuity conditions for some general U-processes. His approach is technically quite involved, and the sufficient conditions given in Ichimura can be difficult to check for a specific nonparametric estimator or a test statistic. In this paper we suggest a simple condition (from Billingsley (1999), see also Mammen (1992)) to establish stochastic equicontinuity of a general stochastic process. The proof of our approach only involves some low-order moment calculation of a stochastic process indexed by a set parameters (taking values in a bounded set). Therefore, in practice it is much easier to check the conditions given in our paper than those given in Ichimura. Moreover, we allow for general weakly dependent data while Ichimura only considers the independent data case. We demonstrate the usefulness of the method by verifying the conditions for several commonly used nonparametric/semiparametric estimators and test statistics. We show that, under some mild conditions, the asymptotic distributions of these estimators/statistics remain unchanged when one replaces the deterministic smoothing parameters by the stochastic counterparts. The paper is organized as follows. In Section 2 we consider a simple univariate density estimation problem to motivate and illustrate our general approach of dealing with dependent smoothing parameters in nonparametric estimation. The main results are presented in Section 3. Some nonparametric estimation and testing examples are given in Section 4. Section 5 presents our Monte Carlo simulations. Section 6 concludes the paper. The Appendix provides the omitted proofs of Section 3.

In this paper we provide a simple sufficient condition to ensure the first order asymptotic equivalence of a nonparametric estimator or a test statistic using stochastic smoothing parameters to that using deterministic smoothing parameters. We provide several examples to illustrate the easiness of checking the condition in practice. One can go on to add more examples, such as the average derivative estimator proposed by Powell et al. (1989), the Smooth Maximum Score estimator by Horowitz (1992), and the single index model estimator by Ichimura (1993), among others discussed in Ichimura (2000). With the results in this paper, one can show that the asymptotic distributions of all these nonparametric/semiparametric estimators remain the same when one replaces the deterministic smoothing parameters by stochastic smoothing parameters provided that View the MathML source|hˆ−h0|=op(|h0|), where h0h0 is a vector of non-stochastic smoothing parameters from the underlined model and View the MathML sourcehˆ is a vector of data-driven selected (stochastic) smoothing parameters. In this paper we only focus on the nonparametric kernel estimation method. It would be of great interest to extend the results to other nonparametric estimation methods such as the nonparametric series estimation (e.g., Chen and Shen (1998), Hahn (1998), Newey (1997)). We leave this possible extension as a future research topic.