時間數列模型之統計推論

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楊巧意(Chiao-Yi Yang) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

時間數列模型之統計推論
(Statistical Inference in Time Series Models)

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摘要(中)

本論文由兩篇獨立的文章所構成，它們探討不同的主題，但皆以時間數列分析為基礎。在第一篇文章中，我們討論非負一階自我迴歸過程之未來值預測問題。首先，不論數列為恆定或具單根，我們證明了一階自我迴歸係數之極值估計式的動差上界和極限分佈。藉由這些理論結果，我們成功地推導了根據極值估計式所得到的最小比例預測式之均方預測誤差(MSPE)的漸近展開式；有了此漸近展開式，我們可以從MSPE的角度，比較此預測式與最小平方預測式的預測表現。比較分析後，我們發現這兩個預測式之MSPE的排序不僅和模型是否存在單根有關，同時也和誤差在零附近之分佈假設有關；因此，若要以MSPE選取具有較佳預測能力的預測式，則有其實務應用上困難之處。為了克服此難題，我們提出的方法是比較這兩個預測式的累積預測誤差(APE)，然後選取具有最小化累積預測誤差值的預測式為最終預測式，並且也證明了此方法所找到之最終預測式漸近等價於具有較小MSPE的預測式。

第二篇文章是在誤差項具有序列相關之線性迴歸模型的假設下，以廣義最小平方(GLS)法估計模型未知參數，其主要目標為提出一個GLS參數估計式，其有效性漸近等價於最佳線性不偏估計式(BLUE)之有效性。為達此目的，建構一個具有一致性的誤差項之共變異矩陣的逆矩陣估計式為重要的關鍵因素。在此文章中，我們藉由修正的Cholesky分解法直接估計共變異矩陣的逆矩陣，而非先估計共變異矩陣，再求取其逆矩陣估計值。此方法所獲得之逆矩陣估計式，在適當的條件之下，不僅具有一致性，同時也保有正定性。而且，根據此逆矩陣估計式所得到的GLS估計式亦具有漸近最佳性質。

在這兩篇文章中，我們透過模擬研究驗證了上述的理論性質，並且實際資料也說明了所提出之方法在實務應用上的可行性。

摘要(英)

This dissertation consists of two self-contained articles
that address two important topics in time series. The first article is concerned with the problem of forecasting a non-negative first-order autoregressive (AR(1)) process. In both the stationary and unit root cases, the moment bounds and limiting distributions of an extreme value estimator of the AR coefficient are established. These results enable us to provide an asymptotic expression for the mean squared prediction error (MSPE) of the minimum ratio predictor,
which is constructed through the extreme value estimator. Based on this expression, we compare the performances of the minimum ratio predictor and the least squares predictor from the MSPE point of view. Our comparison reveals that the better predictor between these two predictors is determined not only by whether a unit root exists or not, but also by the behavior of the underlying error distribution near the origin, and hence is difficult to be
identified in practice. To circumvent this difficulty, we suggest choosing the predictor with the smaller accumulated prediction error (APE) and show that the predictor chosen in this way is asymptotically equivalent to the better one.

The second article provides a method for estimating the model coefficients in the linear regression model with serially correlated errors. The main aim is to propose a generalized least squares (GLS) estimator being as efficient asymptotically as the best linear unbiased estimator (BLUE). To this end, a consistent estimator of
the inverse of the autocovariance matrix of errors is equired. In order to ensure the positive definiteness of the estimated autocovariance matrix, we build an estimator for the inverse of the autocovariance matrix by the use of the modified Cholesky decomposition instead of directly estimating the autocovariance matrix. The resulting matrix estimator converges to the corresponding population matrix at a suitable rate. Moreover, the asymptotic optimality of the corresponding GLS estimator is established.

In these two articles, simulation studies are given to confirm our theoretical finding. In addition, real data analysis is included to illustrate the applicability of proposed methods.

關鍵字(中)

★ 非負一階自我回歸過程
★ 極值估計式
★ 均方預測誤差
★ 累積預測誤差
★ 廣義最小平方估計式
★ 共變異矩陣
★ Cholesky分解法

關鍵字(英)

★ non-negative autoregressive process
★ extreme value estimator
★ mean squared prediction error
★ accumulated prediction error
★ generalized least squares estimator
★ covariance matrix
★ Cholesky decomposition

論文目次

I Predictor Selection for Non-Negative AR(1) Model 1
1 Introduction 2
1.1 Background and Literature Review . . . . . . . . . . . . . . . . . . . . 2
1.2 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Asymptotic Properties of Estimators for the AR coefficients 6
2.1 The Extreme Value Estimator . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The Least Squares Estimator . . . . . . . . . . . . . . . . . . . . . . . 7
3 Mean Square Prediction Error 9
3.1 The MSPE of the Minimum Ratio Predictor . . . . . . . . . . . . . . . 9
3.2 The MSPE of the Least Squares Predictor . . . . . . . . . . . . . . . . 13
4 Predictor Selection Rules Based on the APE 14
5 Numerical Studies 17
5.1 Finite Sample Performance of the Two Candidate Predictors . . . . . . 17
5.2 Finite Sample Performance of Predictor Selection Rules . . . . . . . . . 21
5.3 Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Conclusion and Remarks 27
II Estimation of Linear Regression Models with Serially
Correlated Errors 29
7 Introduction 30
7.1 Background and Literature Review . . . . . . . . . . . . . . . . . . . . 30
7.2 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8 The GLS Estimator of Regression Coefficients 33
8.1 Model and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.2 An Estimator of the Inverse of the Autocovariance Matrix . . . . . . . 34
9 Assumptions and Some Asymptotic Properties 37
9.1 Technical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.2 The Asymptotic Optimality of the Proposed GLS Estimator . . . . . . . 38
10 Numerical Studies 40
10.1 Finite Sample Performance of the Autocovariance Matrix Estimator . . 40
10.2 Finite Sample Performance of the Proposed GLS Estimator . . . . . . . 47
10.3 Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11 Conclusion and Remarks 53
Reference 54
Appendices 59
A: Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . 59
B: Proofs of Theorems 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . . . . . 65
C: Proofs of Theorems 4.1 and 4.2 . . . . . . . . . . . . . . . . . . . . . . . 71
D: Proofs of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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指導教授

銀慶剛、鄒宗山

審核日期

2012-10-1

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