向量自迴歸模型的費雪訊息與超調和先驗分布

Abstract

向量自迴歸模型是一種常用在總體經濟及自然科學上的線性統計模型，本文聚焦在該模型的參數估計問題上。在缺乏對模型參數先驗資訊的情況下，無信息先驗分布是經常被考慮的選項，而傑佛瑞先驗分布是最常使用的分布之一。Komaki(1999)提出了在滿足特定條件下，一種超調和先驗分布將存在並在參數估計上有比傑佛瑞先驗分布更好的表現，而Tanaka(2018)成功的計算出了在一維的自迴歸模型上的超調和先驗分布。本篇論文沿襲了相同的思路，將該理論沿用在向量自迴歸模型上，並嘗試去計算出該模型的超調和先驗分布。本文依序介紹了定義超調和先驗分布所需要的背景知識，包含譜密度、費雪信息及信息幾何學，並給出了在向量自迴歸模型的模型流形上，明確的黎曼流形距離，為進一步的計算奠定了幾何基礎。最後我們提出了在自迴歸模型上，維度對流形距離的影響，並總結了計算該模型的超先驗分布時將遇到的困難及其中幾個可能的解決方法。

The vector autoregressive model (VAR) is a common choice when studying macroeconomics and natural science. In this thesis, we focus on estimating the parameters of the VAR model. When estimating without prior knowledge of the parameters, we often apply a non-informative prior, and Jeffreys prior is one of the most common choices. Komaki(1999) proposed that under certain conditions, a superharmonic prior exists and outperforms the estimation of the Jeffreys prior. Tanaka(2018) successfully derive the superharmonic prior for the autoregressive model. Our research applies this approach to the VAR model and aims to calculate the superharmonic prior for the VAR model. In the following thesis, we first introduce the necessary knowledge to define the superharmonic prior, including spectral density matrix, Fisher information, and information geometry. We compute the explicit form of the Riemannian metric of the VAR model manifold and establish the necessary geometry foundation for further computation. To conclude, we highlight the significant differences between the AR and VAR models, the obstacles when calculating the superharmonic prior, and some possible solutions.