非参数级数估计方法的理论和发展
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摘要
本文研究计量经济学模型非参数估计方法之一的级数估计方法,在总结已有研究成果的基础上,对级数估计理论进行了一定的发展。
首先,对于回归模型,独立同分布数据情形级数估计量的渐近性质已经有过很多的研究,而弱相依数据情形下泛函估计量θ|^ =a(g|^)的渐近性质仅仅有Cheng and Shen(1998)对于更广一类估计方法(筛极值估计)研究过,且只针对a(·)是光滑泛函的情形给出的了高等级条件。对非光滑泛函(比如固定收入水平下的消费者剩余是需求函数的非光滑泛函)还没有相应的研究结果公诸于众。本文将针对严格平稳ρ-混合数据,对于级数估计量的均方收敛速度和一致收敛速度、非光滑泛函的渐近正态性和渐近方差估计量的一致性以及光滑泛函的n~(1/2)-一致性给出原始性条件以及证明。
其次,应用非参数级数估计方法可进行参数模型的设定检验,针对i.i.d.数据,Hong and white (1995)提出了两个一致的单边检验,这两个检验统计量都具有很好的效能并且在某些Monte Carlo模拟研究下表现良好。但是,这些良好的性质依赖于一个较强的假设,那就是零假设下,逼近函数序列必须能够足够快地逼近回归函数。本文提出了一个替代的检验统计量,它具有同样的渐近性质,却不需要前面所提到的假设作为前提。相比Hong and white (1995)的统计量,当回归函数波动较小时,二者具有相似的有限样本性质,但如果回归函数震荡剧烈,本文所构造的统计量明显具有较好的表现。
This dissertation focuses on nonparametric series estimation methods. After summing up the existing results in the literature, we propose two developments on the theory of series methods.
First, the asymptotic properties of the series estimator on the functional of the conditional mean,θ|^ =a(g|^), for independently and identically distributed data have been studied a lot in the literature. The corresponding properties ofθ|^ for weakly dependent data have been derived only for the case where a(·) is a smooth functional (Chen and Shen (1998)) under some high level conditions. The asymptotic properties of the estimatorθ|^ for weakly dependent data for the case where a(·) is not a smooth functional, however, have not been reported in published papers. This dissertation will compute the convergence rates and derive the asymptotic normality ofθ|^ for stationary andρ-mixing data and for any functional a(·) . Our derivations are similar to those in Newey (1997) but ours are more complicated because of data dependence.
Second, through estimating the nonparametric model by series regression, Hong and White (1995) proposed two consistent one-sided test statistics for i.i.d data which are showed to have good power and perform quite well in some situations by a Monte Carlo study. However, the desired asymptotic properties of these test statistics depend on a strong assumption that the underling regression function can be approached by approximating functions quite fast under the null hypothesis. This dissertation presents an alternative test which has the same asymptotic properties without imposing the assumption mentioned above. The new test statistic behaves like Hong and White’s in finite sample for flat regression functions but much better for fluctuant ones.
首先,对于回归模型,独立同分布数据情形级数估计量的渐近性质已经有过很多的研究,而弱相依数据情形下泛函估计量θ|^ =a(g|^)的渐近性质仅仅有Cheng and Shen(1998)对于更广一类估计方法(筛极值估计)研究过,且只针对a(·)是光滑泛函的情形给出的了高等级条件。对非光滑泛函(比如固定收入水平下的消费者剩余是需求函数的非光滑泛函)还没有相应的研究结果公诸于众。本文将针对严格平稳ρ-混合数据,对于级数估计量的均方收敛速度和一致收敛速度、非光滑泛函的渐近正态性和渐近方差估计量的一致性以及光滑泛函的n~(1/2)-一致性给出原始性条件以及证明。
其次,应用非参数级数估计方法可进行参数模型的设定检验,针对i.i.d.数据,Hong and white (1995)提出了两个一致的单边检验,这两个检验统计量都具有很好的效能并且在某些Monte Carlo模拟研究下表现良好。但是,这些良好的性质依赖于一个较强的假设,那就是零假设下,逼近函数序列必须能够足够快地逼近回归函数。本文提出了一个替代的检验统计量,它具有同样的渐近性质,却不需要前面所提到的假设作为前提。相比Hong and white (1995)的统计量,当回归函数波动较小时,二者具有相似的有限样本性质,但如果回归函数震荡剧烈,本文所构造的统计量明显具有较好的表现。
This dissertation focuses on nonparametric series estimation methods. After summing up the existing results in the literature, we propose two developments on the theory of series methods.
First, the asymptotic properties of the series estimator on the functional of the conditional mean,θ|^ =a(g|^), for independently and identically distributed data have been studied a lot in the literature. The corresponding properties ofθ|^ for weakly dependent data have been derived only for the case where a(·) is a smooth functional (Chen and Shen (1998)) under some high level conditions. The asymptotic properties of the estimatorθ|^ for weakly dependent data for the case where a(·) is not a smooth functional, however, have not been reported in published papers. This dissertation will compute the convergence rates and derive the asymptotic normality ofθ|^ for stationary andρ-mixing data and for any functional a(·) . Our derivations are similar to those in Newey (1997) but ours are more complicated because of data dependence.
Second, through estimating the nonparametric model by series regression, Hong and White (1995) proposed two consistent one-sided test statistics for i.i.d data which are showed to have good power and perform quite well in some situations by a Monte Carlo study. However, the desired asymptotic properties of these test statistics depend on a strong assumption that the underling regression function can be approached by approximating functions quite fast under the null hypothesis. This dissertation presents an alternative test which has the same asymptotic properties without imposing the assumption mentioned above. The new test statistic behaves like Hong and White’s in finite sample for flat regression functions but much better for fluctuant ones.
引文
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[3] Ai C R, Sappington D. The impact of the state incentive regulations on the U.S. telecommunication industry. Journal of Regulatory Economics, 22, 2002, 133-160.
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[9] Andrews D W K. Nonparametric kernel estimation for semiparametric models. Econometric Theory, 11, 1995: 560–596.
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[21] Chamberlain G. Efficiency bound for semiparametric regression. Econometrica, 60, 1992:567-596.
[22] Charlier E, Melenberg B, van Soest A. A smoothed maximum score estimator for the binary choice panel data model with application to labor force participation. Statistica Nederlandica, 49, 1995: 324-342.
[23] Chen X, Conley T. A new semiparametric spatial model for panal time series. Journal of Econometrics, 105, 2001:59-83.
[24] Chen X, Fan Y. Consistent hypothesis testing semiparametric and nonparametric models for econometric time series. Journal of Econometrics, 91, 1999: 373-401.
[25] Chen X H, Shen X T. Sieve Extreme Estimates for Weakly Dependent Data. Econometrica 66, 1998: 289-314.
[26] Chen X, White H L. Central limit and functional central limit theorems for Hilbert-valued dependent heterogeneous arrays with applications. Econometric Theory, 14, 1998: 260-284.
[27] Chui C K. An introduction to wavelets. Academic Press, 1992.
[28] Cox D D. Approximation of Least Squares Rerression on Nested Subspaces. Annals of Statistics, 16, 1988: 713-732.
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