非參數模型檢定問題

Abstract

隨機邊界模型經常被使用在生產或成本函數分析。傳統的模型估計方法多半以最大化概似函數估計為主，而建構概似函數需要關於模型中非效率項的分配假設，因此此假設於分析中非常重要。過往的研究文獻中已有若干關於檢定此非效率項分配假設的統計檢定方法，然而其中多半需假設隨機邊界函數為特定的參數化函數。本篇論文中介紹Cheng (2015)的檢定方法並不需要此特別的參數化假設。其主要的概念為利用由非參數估計與半參數估計所得出的殘差項導出之經驗非配函數的差，建構出Kolmogorov-Smirnov 檢定統計量以進行檢定。若非效率項的真實條件分配確實如預先設定的假定分配，由非參數估計與半參數估計所得出的殘差項之分配應約略相同，此為本文中統計方法的立論概念。由於在虛無假設下漸進分配函數的複雜性，我們使用重複抽樣法(bootstrap method)來得出檢定所需的p -值。模擬的數值分析結果顯示此方法有不錯的效果量與檢定力。

Stochastic frontier model is widely used in studying production or cost frontiers. Traditional approach for the estimation of parametric stochastic frontier function is to maximize the constructed log-likelihood to get the estimators of the model parameters. The construction of the log-likelihood requires distributional assumptions of the inefficiency term. Therefore, the performance of the frontier model estimation relies heavily on the accuracy of this distributional specification of the inefficiency term. There have been several testing procedures in the literature dealing with this kind of specification problem, under the assumption of parametric stochastic frontier function. While the literature focus on modeling, inference and testing specification of the (unobserved) inefficiency term, the problems are always coupled with specification of the frontier function. In other words, validity of the analysis of the inefficiency is dependent on assumed parametric form of the frontier function. We investigate the emerging issue of testing some parametric specification of the conditional distribution of the inefficiency given the covariate, without parametric assumption on the frontier function. Existing methods uses information on specifications of both components. Hence, the null hypothesis is true only if both specifications are correct and when it is rejected there is no clue which specification is violated relies heavily on the accuracy of this distributional specification of the inefficiency term. There have been several testing procedures in the literature dealing with this kind of specification problem, under the assumption of parametric stochastic frontier function. While the literature focus on modeling, inference and testing specification of the (unobserved) inefficiency term, the problems are always coupled with specification of the frontier function. In other words, validity of the analysis of the inefficiency is dependent on assumed parametric form of the frontier function. We investigate the emerging issue of testing some parametric specification of the conditional distribution of the inefficiency given the covariate, without parametric assumption on the frontier function. Existing methods uses information on specifications of both components. Hence, the null hypothesis is true only if both specifications are correct and when it is rejected there is no clue which specification is violated. The main idea of the proposed specification test in this thesis is to construct the Kolmogorov-Smirnov test statistic via using the difference between the two empiri- cal distributions of the residuals from nonparametric and semiparametric estimation. Without the distributional assumption of the inefficiency term, we may still esti- mate the frontier function via using nonparametric techniques, which results the first version of the wanted residuals. With the imposed parametric specification of the conditional distribution of the inefficiency , we can utilize the relationship between the conditional moment of the residuals and the parameters of the conditional distri- bution of the inefficiency to estimate the frontier function, which results the second version of the residuals. The rationale here is, if the parametric specification of the conditional distribution of the inefficiency is true, the residuals obtained from fully nonparametric estimation and the residuals obtained from semiparametric estimation should roughly have the same distributions. This idea forms the basis of the testing procedure. Because of the complexity of the asymptotic null distribution, we employ bootstrap to generate the p-values. We examine numerical performance of this non- parametric approach to test the specification of the inefficiency proposed by Cheng (2015) via an extensive simulation study. Our simulation study includes two sets of the parametric specification of the conditional distribution of the inefficiency , one is the standard half-normal distribution setting, and the other is the log-normal distri- bution. Heteroscedasticity in the conditional distribution of the inefficiency term is also considered. We find that the test has good level accuracy and nontrivial power even under heteroscedasticity.