不需估計共變異結構特徵組的新平滑性檢定

Abstract

假設檢定在經濟、財務或是計量分析中都是相當重要的課題。一般而言，研究者通常會先選擇使用綜合性的一致性檢定 (omnibus consistent test)。儘管這些綜合性檢定在理論上可以檢測任何背離虛無假設的情況，然而，藉由頻譜分析 (spectral analysis)，文獻上已知綜合性檢定會對「高頻(high frequency)」對立假設缺乏檢定力。為了避免此問題，已有許多學者提出各式相對應的平滑性檢定 (smooth tests)，而且其良好的理論和實證性質都也已被揭露。基本上，現行的平滑性檢定架構主要是建立在資料序列間的共變異結構 (covariance structure)所對應的特徵組 (eigen-pairs)上。然而，文獻上僅有極少數的資料序列所對應的特徵組有明確的型式可供分析使用。為了改善此一缺失，在此計畫中，我提出了一個不需要資料序列間的共變異結構即可建立統計量的新平滑性檢定架構。在給定任何基本函數 (basis functions)下，我們可以有相對應於資料序列的傅立葉表現式 (Fourier representation) 及傅立葉係數 (Fourier coefficients)。基本上，這個新的平滑性檢定架構就是建立在這些傅立葉係數標準化後的主成分 (normalized principal components)上。在這樣的架構下，我們進而提出兩個由資料所驅策 (data-driven) 的新平滑性檢定。簡而言之，與目前文獻上常用的平滑性檢定相比，這個計畫提出了一個具一般化且容易執行的新平滑性檢定架構來改善綜合性檢定所可能面臨之缺乏檢定力的問題。我們的模擬結果也支持這樣的方法。

As is well known,hypothesis testing is an important task in economic, financial or econometric analysis.Generally speaking,considering an omnibus consistent test is usually the first attempt for many researchers when there is no particular preferred alternative to the null hypothesis.Despite the capability of an omnibus test to detect deviations in all directions, it is also well known that the omnibus tests may lack power against the so-called ""high frequency‘‘ alternatives after invoking spectral analysis. To avoid this ""power deficiency‘‘ problem in the omnibus tests, smooth tests are then proposed in various issues hypothesis testing problems, and good theoretical properties and empirical evidence of smooth tests have been documented by many researchers. In the existing smooth tests framework, the eigen-pairs of the (asymptotic) covariance structure of the processplays the central role. However, only for some few processes, we may have explicit forms for the corresponding eigen-pairs. In order to remedy this deficiency, this project aims to propose more general and more implementable smooth-type test. In this project,we propose a new smooth-type test approach without any knowledge of the covariance structure of the process in forming the associated test statistic. Given any basis functions, we have Fourier representations for the process and the corresponding Fourier coefficients.In essence, this new smooth-type test approach is established by using the normalized principal components for these Fourier coefficients. Moreover, two associated data-driven smooth-type tests are also proposed. In sum, other than the conventional smooth tests,this project provide a new, general and easy-to-implement approach to increasing thetesting powers of the omnibus tests. The simulation results support this approach.