半参数变系数部分线性度量误差模型中的序列相关检验和经验似然
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摘要
半参数变系数部分线性模型是近年来才提出的一个内容丰富,应用广泛的新模型。该模型包括了许多通常的参数,半参数和非参数回归模型。线性回归模型,部分线性回归模型和变系数模型都是该模型的退化情形。和参数线性模型或者半参数部分线性模型相比较,半参数变系数部分线性模型允许更灵活的函数形式,同时避免了许多“维数祸根”问题。
在经典的回归模型中,一般假定误差项是相互独立的,且具有相同方差的白噪声。如果独立性假设破坏了,则模型存在序列相关;若方差不等,则称模型存在异方差。若模型存在序列相关性,则会导致如下问题:参数估计量非有效;变量的显著性检验失去意义,其他检验也是如此;模型的预测失效;可能忽略了某些重要的解释变量,甚至是模型被误用等。若模型存在异方差性,也会存在同样的问题。因此,在统计推断之前,检验模型是否存在序列相关和异方差很有必要。
多年来,人们已经认识到检验线性回归模型中误差项是否存在序列相关的重要性,并且对这个问题进行了广泛深入的研究。而对非参数和半参数回归模型中的序列相关检验研究较少,直到最近才有人开始研究多元回归模型,部分线性模型,有界非参数函数,半参数时序模型和半参数panel数据模型的序列相关检验。至于近年来提出的半参数变系数部分线性模型和半参数变系数部分线性时序模型,其序列相关检验至今鲜有报道。因此,本文在方差齐性的假设下,详细研究了这些模型的序列相关检验问题,得出了一些有益的结果。
在实际应用中,由于人为或系统的原因,度量误差总是存在的,因此,研究度量误差模型具有更大的实用价值。线性度量误差模型和部分线性度量误差模型的估计和性质已被广泛研究,而半参数变系数部分线性度量误差模型的研究尚未引起重视。目前关于这个模型的研究主要是You和Chen(2006)的论文。本文以You和Chen(2006)提出的估计和性质为基础,在线性部分具有可加度量误差和误差协差阵已知(识别条件一)的情况下探讨了半参数变系数部分线性度量误差模型中的序列相关检验问题;接着在度量误差的协差阵和模型误差方差的比值已知(识别条件二)的条件下,得到了半参数变系数部分线性度量误差模型的参数估计和非参数估计及其大样本性质,并在此基础上研究了该模型的序列相关检验问题。
关于上述模型中的异方差检验,本文尚未考虑,但这个课题值得深入研究。
经验似然是Owen(1988,1990)提出的一种非参数统计推断方法。这一方法与经典或现代的统计方法比较有很多突出的优点,如:用经验似然方法构造置信区间有域保持性、变换不变性及置信域的形状由数据自行决定等诸多优点。正因为如此,这一方法引起了许多统计学家的兴趣,他们将这一方法应用到各种统计模型及各种领域。本文提出了用经验对数似然比来检验半参数变系数部分线性模型及其度量误差模型中的序列相关性,并且在测量数据具有可加误差的情况下考虑了其度量误差模型中的经验似然推断。此外,本文也研究了纵向的半参数变系数部分线性模型的块经验似然推断。
本文的研究成果主要有以下五个:第一,把经验似然引入半参数变系数部分线性模型的序列相关检验中,提出了经验对数似然比检验;第二,把经验似然引入含度量误差的半参数变系数部分线性模型的序列相关检验中,在识别条件一和识别条件二下分别得到了经验对数似然比的非参数Wilks'定理;第三,把Li和Hsiao(1998)的方法推广到了半变系数部分线性模型、半参数变系数部分线性度量误差模型、半参数变系数部分线性时序模型和半参数变系数部分线性panel数据模型中来检验序列相关;第四,把经验似然应用到半参数变系数部分线性度量误差模型中,丰富和发展了经验似然理论;第五,把块经验似然应用到纵向的半参数变系数部分线性模型中,得到了所提块经验对数似然比的渐近卡方分布。
Semiparametric varying-coefficient partially linear model, which has recently been proposed as a new model, holds ample contents and extensive applications. It includes many usual parametric, semiparametric and nonparametric regression models. Linear regression model, partially linear regression model and varying-coefficient model are its degeneration cases. In comparison with parametric linear model or semiparametric partially linear models, semiparametric varying-coefficie -nt partially linear model allows more flexible function forms, and avoids many "curse of dimensionality" problems. In the context of classic regression models, we generally assume that the errors are mutually independent and homoscedastic. If the errors are not independent, we say that the models are serially correlated. If the error variances are not equal, we say that the models are heteroscedastic. If the errors are serially correlated, we will face with the following problems: the parameter estimate is not efficient; the significant test for variable is meaningless, and the same to other tests; the forecast for the model is inefficient; some important explanatory variables are omitted or the functional form is misspecified. When the model is heteroscedastic, we can face with the same problems as serial correlation. Therefore, it is necessary to test the presence of serial correlation and heteroscedasticity before statistic inference.
The importance of testing for serial correlation in the error terms of a linear regression model has been recognized for many years. Some authors have investigated in detail this problem in the linear regression model. As far as I know, no one has tested for serial correlation in semiparametric and nonparametric regression models. Until recently some authors have begun testing for serial correlation in multivariate regression models, partial linear models, bounded nonparametric functions, semiparametric time series models and semiparametric panel data models. As regards semiparametric varying-coefficient partially linear models and semiparametric varying-coefficient partially linear time series models which are recently proposed, no one has investigated serial correlation test in these models. Hence, in this paper, we particularly investigate serial correlation test in these models, and obtain some useful results.
In many applications, however, there often exist measurement errors because of the nature of measurement mechanism or man-made factor. So it is more practical to investigate the measurement error (errors-in-variables) models than the ordinary regression models. Estimators and properties on linear errors-in-variables models and partially linear errors-in-variables models have been extensively investigated, while the research on semiparametric varying-coefficient partially linear errors-in-variables model hasn't attached enough importance. To this day, the research on this model has mainly the paper of You and Chen(2006). Based on estimators and properties proposed by You and Chen(2006) in semiparametric varying-coefficient partially linear errors-in-variables model, in this paper we particularly investigate serial correlation test under the case where the covariates are measured with additive errors and the covariance of measurement error is known(i.e., the identifying condition 1). Then, we construct the estimators of the parameter components and nonparametric components, and obtain their asymptotic properties under the case where the covariates are measured with additive errors and the variance ratio of measurement error to regression equation error is assumed to be known(i.e., the identifying condition 2). Based on these estimators and properties, we investigate serial correlation test in the model.
However, in this paper we don't investigate testing heteroscedasticity in the aforementioned models. But this is certainly worthy of effort for further research.
Empirical likelihood, proposed by Owen(1988, 1990), is a nonparametric method of inference. Compared with other classic or modern statistic methods, empirical likelihood has many prominent merits. For example, the empirical likelihood ratio confidence region is range preserving and transformation respecting, and the shape and orientation of the resulting confidence regions are determined entirely by the data. Therefore, the empirical likelihood has aroused many statisticians' interest, and been applied to many fields and statistic models. In this paper we propose an empirical log-likelihood ratio to testing serial correlation in semiparametric varying-coefficient partially linear model and semiparametric varying-coefficient partially linear errors-in-variables model, and investigate empirical likelihood inference in semiparametric varying-coefficient partially linear errors-in-variables model under the case where the measurement error is assumed to be additive. Besides, we also consider the block empirical likelihood inference in the longitudinal semiparametric varying-coefficient partially linear model.
This paper obtains the following five main results: the first result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear models and propose the empirical log-likelihood ratio test. The second result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear errors-in-variables models and obtain the nonparametric version of Wilks' theorem of the proposed empirical log-likelihood ratio statistics under the identifying condition 1 and 2, respectively. The third result is that we generalize the method proposed by Li and Hsiao(1998) to semiparametric varying-coefficient partially linear models, semiparametric varying-coefficient partially linear errors-in-variables model, semiparametric varying-coefficient partially linear time series models and semiparametric varying-coefficient partially linear panel data models. The fourth result is that we apply empirical likelihood to semiparametric varying-coefficient partially linear errors-in-variables models, enrich and develop the empirical likelihood theory. The fifth result is that we apply the block empirical likelihood to the longitudinal semiparametric varying-coefficient partially linear models, and obtain the chi-square limiting distribution of the proposed block empirical log-likelihood ratio.
在经典的回归模型中,一般假定误差项是相互独立的,且具有相同方差的白噪声。如果独立性假设破坏了,则模型存在序列相关;若方差不等,则称模型存在异方差。若模型存在序列相关性,则会导致如下问题:参数估计量非有效;变量的显著性检验失去意义,其他检验也是如此;模型的预测失效;可能忽略了某些重要的解释变量,甚至是模型被误用等。若模型存在异方差性,也会存在同样的问题。因此,在统计推断之前,检验模型是否存在序列相关和异方差很有必要。
多年来,人们已经认识到检验线性回归模型中误差项是否存在序列相关的重要性,并且对这个问题进行了广泛深入的研究。而对非参数和半参数回归模型中的序列相关检验研究较少,直到最近才有人开始研究多元回归模型,部分线性模型,有界非参数函数,半参数时序模型和半参数panel数据模型的序列相关检验。至于近年来提出的半参数变系数部分线性模型和半参数变系数部分线性时序模型,其序列相关检验至今鲜有报道。因此,本文在方差齐性的假设下,详细研究了这些模型的序列相关检验问题,得出了一些有益的结果。
在实际应用中,由于人为或系统的原因,度量误差总是存在的,因此,研究度量误差模型具有更大的实用价值。线性度量误差模型和部分线性度量误差模型的估计和性质已被广泛研究,而半参数变系数部分线性度量误差模型的研究尚未引起重视。目前关于这个模型的研究主要是You和Chen(2006)的论文。本文以You和Chen(2006)提出的估计和性质为基础,在线性部分具有可加度量误差和误差协差阵已知(识别条件一)的情况下探讨了半参数变系数部分线性度量误差模型中的序列相关检验问题;接着在度量误差的协差阵和模型误差方差的比值已知(识别条件二)的条件下,得到了半参数变系数部分线性度量误差模型的参数估计和非参数估计及其大样本性质,并在此基础上研究了该模型的序列相关检验问题。
关于上述模型中的异方差检验,本文尚未考虑,但这个课题值得深入研究。
经验似然是Owen(1988,1990)提出的一种非参数统计推断方法。这一方法与经典或现代的统计方法比较有很多突出的优点,如:用经验似然方法构造置信区间有域保持性、变换不变性及置信域的形状由数据自行决定等诸多优点。正因为如此,这一方法引起了许多统计学家的兴趣,他们将这一方法应用到各种统计模型及各种领域。本文提出了用经验对数似然比来检验半参数变系数部分线性模型及其度量误差模型中的序列相关性,并且在测量数据具有可加误差的情况下考虑了其度量误差模型中的经验似然推断。此外,本文也研究了纵向的半参数变系数部分线性模型的块经验似然推断。
本文的研究成果主要有以下五个:第一,把经验似然引入半参数变系数部分线性模型的序列相关检验中,提出了经验对数似然比检验;第二,把经验似然引入含度量误差的半参数变系数部分线性模型的序列相关检验中,在识别条件一和识别条件二下分别得到了经验对数似然比的非参数Wilks'定理;第三,把Li和Hsiao(1998)的方法推广到了半变系数部分线性模型、半参数变系数部分线性度量误差模型、半参数变系数部分线性时序模型和半参数变系数部分线性panel数据模型中来检验序列相关;第四,把经验似然应用到半参数变系数部分线性度量误差模型中,丰富和发展了经验似然理论;第五,把块经验似然应用到纵向的半参数变系数部分线性模型中,得到了所提块经验对数似然比的渐近卡方分布。
Semiparametric varying-coefficient partially linear model, which has recently been proposed as a new model, holds ample contents and extensive applications. It includes many usual parametric, semiparametric and nonparametric regression models. Linear regression model, partially linear regression model and varying-coefficient model are its degeneration cases. In comparison with parametric linear model or semiparametric partially linear models, semiparametric varying-coefficie -nt partially linear model allows more flexible function forms, and avoids many "curse of dimensionality" problems. In the context of classic regression models, we generally assume that the errors are mutually independent and homoscedastic. If the errors are not independent, we say that the models are serially correlated. If the error variances are not equal, we say that the models are heteroscedastic. If the errors are serially correlated, we will face with the following problems: the parameter estimate is not efficient; the significant test for variable is meaningless, and the same to other tests; the forecast for the model is inefficient; some important explanatory variables are omitted or the functional form is misspecified. When the model is heteroscedastic, we can face with the same problems as serial correlation. Therefore, it is necessary to test the presence of serial correlation and heteroscedasticity before statistic inference.
The importance of testing for serial correlation in the error terms of a linear regression model has been recognized for many years. Some authors have investigated in detail this problem in the linear regression model. As far as I know, no one has tested for serial correlation in semiparametric and nonparametric regression models. Until recently some authors have begun testing for serial correlation in multivariate regression models, partial linear models, bounded nonparametric functions, semiparametric time series models and semiparametric panel data models. As regards semiparametric varying-coefficient partially linear models and semiparametric varying-coefficient partially linear time series models which are recently proposed, no one has investigated serial correlation test in these models. Hence, in this paper, we particularly investigate serial correlation test in these models, and obtain some useful results.
In many applications, however, there often exist measurement errors because of the nature of measurement mechanism or man-made factor. So it is more practical to investigate the measurement error (errors-in-variables) models than the ordinary regression models. Estimators and properties on linear errors-in-variables models and partially linear errors-in-variables models have been extensively investigated, while the research on semiparametric varying-coefficient partially linear errors-in-variables model hasn't attached enough importance. To this day, the research on this model has mainly the paper of You and Chen(2006). Based on estimators and properties proposed by You and Chen(2006) in semiparametric varying-coefficient partially linear errors-in-variables model, in this paper we particularly investigate serial correlation test under the case where the covariates are measured with additive errors and the covariance of measurement error is known(i.e., the identifying condition 1). Then, we construct the estimators of the parameter components and nonparametric components, and obtain their asymptotic properties under the case where the covariates are measured with additive errors and the variance ratio of measurement error to regression equation error is assumed to be known(i.e., the identifying condition 2). Based on these estimators and properties, we investigate serial correlation test in the model.
However, in this paper we don't investigate testing heteroscedasticity in the aforementioned models. But this is certainly worthy of effort for further research.
Empirical likelihood, proposed by Owen(1988, 1990), is a nonparametric method of inference. Compared with other classic or modern statistic methods, empirical likelihood has many prominent merits. For example, the empirical likelihood ratio confidence region is range preserving and transformation respecting, and the shape and orientation of the resulting confidence regions are determined entirely by the data. Therefore, the empirical likelihood has aroused many statisticians' interest, and been applied to many fields and statistic models. In this paper we propose an empirical log-likelihood ratio to testing serial correlation in semiparametric varying-coefficient partially linear model and semiparametric varying-coefficient partially linear errors-in-variables model, and investigate empirical likelihood inference in semiparametric varying-coefficient partially linear errors-in-variables model under the case where the measurement error is assumed to be additive. Besides, we also consider the block empirical likelihood inference in the longitudinal semiparametric varying-coefficient partially linear model.
This paper obtains the following five main results: the first result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear models and propose the empirical log-likelihood ratio test. The second result is that we introduce empirical likelihood to test serial correlation in semiparametric varying-coefficient partially linear errors-in-variables models and obtain the nonparametric version of Wilks' theorem of the proposed empirical log-likelihood ratio statistics under the identifying condition 1 and 2, respectively. The third result is that we generalize the method proposed by Li and Hsiao(1998) to semiparametric varying-coefficient partially linear models, semiparametric varying-coefficient partially linear errors-in-variables model, semiparametric varying-coefficient partially linear time series models and semiparametric varying-coefficient partially linear panel data models. The fourth result is that we apply empirical likelihood to semiparametric varying-coefficient partially linear errors-in-variables models, enrich and develop the empirical likelihood theory. The fifth result is that we apply the block empirical likelihood to the longitudinal semiparametric varying-coefficient partially linear models, and obtain the chi-square limiting distribution of the proposed block empirical log-likelihood ratio.
引文
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