閾值擴散過程的近似最大概似估計法

Abstract

本論文係根據Su和Chan於2015年~cite{su2015quasi}所提出的quasi-likelihood estimator(QLE)所發想而成，其中我們關注在如何計算單一閾值擴散過程(a threshold diffusion process)其最大概似估計量(maximum likelihood estimator)的問題。起因是現實世界的觀測的資料是離散而且可能是不規則區間(irregularly-spaced)，且針對閾值擴散過程其最大概似估計量(maximum likelihood estimator)也因其概似函數(likelihood function)為非線性的結構，所以閾值擴散過程的最大概似估計量只有以隨機積分(stochastic integrals)表示的隱形式(implicit form)，也因此產生一個問題：如何使用離散而不規則的資料去“近似”單一閾值擴散過程的最大概似估計量？針對這個問題，我們提出了所謂“近似最大概似估計法(approximate maximum likelihood method)”去估計單一閾值擴散過程上的參數，而根據此法而得的估計量則稱為“近似最大概似估計量(approximate maximum likelihood estimator；AMLE)”；更進一步，我們利用模擬的結果去給出近似最大概似估計量的大樣本性質，並且也利用這個方法針對長期的利率結構進行一些判讀，而使用的利率資料為Federal Reserve Bank's H15 資料集中的three-month US treasury rate 和 10-year treasury constant maturity rate-3-month treasury bill: secondary market rate 。

Based on the idea of quasi-likelihood estimator(QLE) in Su and Chan(2015), we focus on a problem arisen from estimating the maximum likelihood estimators(MLEs) for a threshold diffusion process. Since the data observed are discrete in the real world, and MLE for a threshold diffusion process is an implicit form of stochastic integrals due to the nonlinear structure of likelihood function of the threshold diffusion process, there might arise the question: how to 'approximate' the MLEs via using the discrete data without the analytic form of the estimator? Therefore, we propose an approximate maximum likelihood method for estimating MLEs of a threshold diffusion process, and the estimator we obtain is called approximate maximum likelihood estimator(AMLE). Moreover, from the simulation results, we give some conjectures about the large sample properties of the AMLE. Finally, we apply our method to study the term structure of a long time series of US interest rates (three-month US treasury rate and 10-year treasury constant maturity rate-3-month treasury bill: secondary market rate, which are based on the Federal Reserve Bank's H15 data set).