線型與非線型時間序列預測誤差的漸進分析

Abstract

本論文考慮自迴歸時間趨勢模型 (以下稱模型一) 與自迴歸自我觸發式門檻模型 (以下稱模型二) 之均方誤差的漸進表達式。關於模型一，我們提出一費雪訊息矩陣 (Fisher information matrix) 等價定理並藉此推導出在非常一般的時間趨勢下的預測均方差。模型二包含了一門檻值與一個門檻延遲項，此二者是決定未來值的關鍵。根據本文計算，模型二的預測除取決於回歸參數的估計值外，亦與門檻延遲值落點有關。舉例而言，當模型二門檻延遲項落於門檻估計值與門檻真值之間時，預測者會誤判模型。此種誤判機率甚低，但一旦發生，將造成極大的預測誤差。本文主要貢獻之一為模型二的預測誤差包含常見的回歸係數估計誤差與模型誤判誤差。正確的均方誤差表達式將有助於模型選擇等統計應用。本文除了預測理論上有所突破外，在模型選擇及矩陣代數的研究上亦帶來新的啟發。

In this work, we provide asymptotic expressions for mean squared prediction error (AMSPE) of autoregressive models with time trend and of self-exciting autoregressive threshold models (SETAR). For time trend models, we provide a characteristic theorem of Fisher information matrix, which in turn is used for deriving AMSPE of AR models with a general time trend. On the other hand, a SETAR process includes a threshold and threshold lag term deciding the state at the next period. According to our calculation, both coefficient and state estimation are crucial to prediction of SETAR models. Misjudgement of state occurs when the estimated state is not the true state. More specifically, it occurs when the value of threshold lag term at current time falls into the interval of estimated threshold and the real one. Such misjudgement happens with low probability, but when it does, it causes much prediction error. Our result shows AMSPE of SETAR includes both estimation variance and misjudgement of the state. Definite results of AMSPE can facilitate many statistical applications such as model selection. On the whole, besides analysis of prediction error, our work contributes to model selection and matrix algebra.