臺灣博碩士論文加值系統

本論文旨在探討三項計量經濟檢定方法的議題，包括非線性與線性架構內，變異數異質性未知函數型式下的統計推論，以及預測有用性檢定。前兩項係探討實証研究上時常遭遇的變異數異質性現象，即樣本觀察值的變異數可能隨解釋變數變化而有差異，致古典假設未被滿足所衍生的推論問題。眾所周知，在線性模型架構下，變異數異質性不會影響估計式的一致性，然而，其效率性與型一誤差掌控等小樣本性質則不盡理想。事實上，該等特質同樣對非線性模型造成困擾，文獻上卻缺乏較細緻的應對之道。因此，我們嘗試發展一些推論程序，期能據以改善現行計量方法的相關缺失。至於預測的議題，我們的主要貢獻在於提出合適的檢定程序，透過延伸Ashley (1983) 提出的預測有用性測度，協助使用者判斷預測數據是否值得引用的問題。

首先關於變異數異質性問題，過去文獻探討的主要課題，是如何改善線性模型下檢定量的小樣本性質。然而計量經濟研究中，非線性設定雖十分普遍，面對變異數異質性卻缺乏較細緻的因應作法。在一致性與效率性等考量下，對於變異數異質性下非線性模型的統計推論，其中一種最廣被運用的效率估計程序，係採取一般化動差法 (generalized method of moment, GMM) 並將共變異矩陣利用殘差平方之對角矩陣取代，即White (1980) 的方式。我們的分析重點在於非線性模式下變異數異質性造成的小樣本問題。較具體地說，在非線性迴歸模式的GMM架構下，我們提出直接延伸傳統HC2-HC5應用範疇的改進版本。透過本研究的蒙地卡羅試驗顯示，當GMM採取非線性模型專用的HCCME (heteroskedasticity consistent covariance matrix estimator) 後，估計與假設檢定的小樣本表現確實能獲得改善。此外，我們也針對非線性模型專用的HCCME，提出貝氏觀點的詮釋。這方面的成果係延伸Poirier (2010) 以貝氏觀點研析線性模型採用HCCME的討論。事實上，透過GMM的架構，該等詮釋法除了非線性模型，尚將工具變數一併納入。

其次在變異數異質性下，研究者雖可藉由普通最小平方法 (ordinary least squares, OLS) 搭配HCCME做出具大樣本效力 (asymptotically valid) 的統計推論，其大樣本效率 (asymptotic efficiency) 卻不如可執行的一般化最小平方法 (feasible generalized least squares, FGLS) 或 Cragg (1983) 估計式。值得一提的是，Flachaire (2005) 結合Cragg估計式與Davidson and Flachaire (2001) 提出的wild bootstrap檢定，分析採用效率估計式的檢定程序。Flachaire (2005) 並透過模擬試驗指出，該等檢定程序具有優良的小樣本性質。有鑑於此，本研究乃提出結合半母數效率估計式與wild bootstrap的檢定程序，並透過蒙地卡羅試驗，對型一誤差的掌控能力與檢定力進行評比，據以分析該等檢定程序的小樣本表現。根據本研究的模擬結果，我們提出的檢定程序具有良好的型一誤差控制力，並具有堪與Flachaire (2005) 的程序相互比擬之檢定力。

關於預測議題方面，Ashley (1983) 提出判定經濟變數之預測資訊是否值得引用的準則，該項指標對於決策者而言甚具參考價值。據此，本研究乃發展三項檢定程序，包括ratio-based檢定、difference-based檢定，以及貝氏方法，據以延伸Ashley (1983) 的有用性準則。其中，貝氏方法的優點，是具有涵蓋決策環境內所有可能訊息的彈性。例如隨著國民經濟統計系統演進，變數定義會隨之發生改變，而改變之前變數間的關係，有時亦可視為一項先驗訊息而納入分析。關於ratio-based與difference-based檢定，我們尚透過模擬試驗，分析它們的小樣本性質，並據以提供相關建議。此外，本研究亦收集《專家預測者調查》 (Survey of Professional Forecasters, SPF) 對六項美國總體經濟變數的預測，以示範本研究提出新方法之應用。準此，研究者得以正式地檢定預測資訊是否確實值得引用。

This dissertation aims to deal with three issues on testing in econometrics: Inference under heteroskedasticity of unknown form for nonlinear and linear models respectively, and usefulness tests of forecasts. The first two discuss a common problem in empirical econometric studies. That is the presence of heteroskedasticity in the error terms - the variance in many cases varies with the regressors, which violates the famed classical assumptions. It is well known that, while heteroskedasticity in linear regression models does not affect the consistency of the underlying estimator, the finite-sample properties and efficiency are not satisfying. We thus try to develop procedures to make some improvements. As for the issue on forecasting, we make a complement to the literature by extending the usefulness measure of forecasts proposed in Ashley (1983) to be a testable hypothesis.

First, previous studies concentrate on improving the finite-sample properties of test in linear models under heteroskedasticity, however, nonlinear specifications are also popular in econometrics. Considering both consistency and efficiency for inference on nonlinear models under heteroskedasticity, one of the most commonly used methods is using an efficient estimation, i.e., GMM, and replacing the covariance matrix with a diagnal matrix using squared residuals just like White (1980). In this aspect, our focus is placed on the finite-sample problems caused by the heteroskedasticity in nonlinear models. Specifically, within the framework of nonlinear models, we propose a straightforward approach by extending the applicability of HC2--HC5 for GMM. The small sample refinements of the estimation and hypothesis testing, using GMM with the modified version of HCCME (heteroskedasticity consistent covariance matrix estimator), are demonstrated via our Monte Carlo experiments. In addition, we also provide Bayesian interpretations for the modified versions of HCCME under the setting of nonlinear model, which is an extension of Poirier's (2010) Bayesian explorations for HCCME from a linear model without instrument variables to the GMM framework allowing nonlinear specification and instrument variables to be considered.

Second, even though we can still make asymptotically valid inference using the ordinay least squares estimator (OLS) with HCCME, it is asymptotically less efficient than some alternative estimators such as feasible generalized least squares estimator (FGLS) or Cragg (1983) estimator. Flachaire (2005) studies the efficient tests robust to heteroskedasticity which is based on the combination of Cragg estimator and wild bootstrap test proposed by Davidson and Flachaire (2001). Motivated by the good finite sample performance from simulation experiments in Flachaire (2005), we would like to explore an alternative class of testing procedures combining the semi-parametric estimators and wild bootstrap test. In this aspect, we study the small sample properties via comparing the size distortions and testing powers of the test statistic. Our simulation experiments provide evidences that the proposed testing procedure may have potential to complement the existing efficient tests.

Regarding the issue of forecasting, Ashley (1983) proposes a criterion (known as Ashley's index) to judge whether the external macroeconomic variables are well forecasted to serve as explanatory variables in forecasting models, which is crucial for policy makers. In this study, we try to extend Ashley's work by providing three testing procedures, including a ratio-based test, a difference-based test, and the Bayesian approach. The Bayesian approach has the advantage of allowing the flexibility of adapting all possible information content within a decision-making environment such as the change of variable's definition due to the evolving system of national accounts. We demonstrate the proposed methods by applying six macroeconomic forecasts in the Survey of Professional Forecasters. Researchers or practitioners can thus formally test whether the external information is helpful.

Abstract vi
Chapter 1: Finite-Sample Refinement of GMM Approach to Nonlinear Models Under
Heteroskedasticity of Unknown Form
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Model and Two-Step GMM Estimator . . . . . . . . . 4
1.3 The Proposed Method . . . . . . . . . . . . . . . . . 6
1.3.1 First Order Expansion for GMM Estimator . . . . . . 6
1.3.2 The Quasi-Hat Matrix . . . . . . . . . . . . . . . . 7
1.3.3 Modified HCCME for Nonlinear Models . . . . . . . . 9
1.3.4 Quasi-t Statistic Using Modified HCCME . . . . . . 10
1.4 Monte Carlo Experiments . . . . . . . . . . . . . . . 11
1.4.1 Simulation Design . . . . . . . . . . . . . . . . . 11
1.4.2 Simulation Results . . . . . . . . . . . . . . . . 12
1.4.3 Further Discussion . . . . . . . . . . . . . . . . 14
1.5 An Empirical Application . . . . . . . . . . . . . . 15
1.5.1 CES Production Function . . . . . . . . . . . . . . 15
1.5.2 Results Using the Original Data . . . . . . . . . . 15
1.5.3 Results Using an Artificial Data . . . . . . . . . 16
1.6 The Bayesian Interpretations . . . . . . . . . . . . 17
1.6.1 Bayesian Bootstrap . . . . . . . . . . . . . . . . 17
1.6.2 Posterior Moments Under GMM Framework . . . . . . . 18
1.6.3 Interpretations . . . . . . . . . . . . . . . . . . . . . 20
1.7 Concluding Remarks . . . . . . . . . . . . . . . . . 21
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 2: Improved Tests for Linear Regression Models With Heteroskedasticity of
Unknown Form
2.1 Introduction . . . . . . . . . . . . . . . . . . . . 48
2.2 The Model and Estimators . . . . . . . . . . . . . . 50
2.3 Proposed Tests Based on Semi-parametric FGLS . . . . 53
2.3.1 Semi-parametric FGLS Estimators . . . . . . . . . . 53
2.3.2 Asymptotic and Bootstrap Tests . . . . . . . . . . 54
2.3.3 Tests With Unrestricted and Restricted Residuals . 56
2.4 Monte Carlo Experiments . . . . . . . . . . . . . . . 57
2.4.1 Simulation Design . . . . . . . . . . . . . . . . . 57
2.4.2 NW-Based FGLS’s ERP and Power Curves . . . . . . . 58
2.4.3 Results from ??NN, Local Linear and Series Estimators . . . . . . . . . . . . . . . . . . . . . . . 60
2.5 Comparison With Cragg-Based Bootstrap Test . . . . . 60
2.5.1 Cragg-Based Bootstrap Test . . . . . . . . . . . . 60
2.5.2 Evaluation via ERP and Power Curves . . . . . . . . 61
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . 62
Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 3: Usefulness Tests of Forecasts
3.1 Introduction . . . . . . . . . . . . . . . . . . . . 84
3.2 Tests for the Usefulness of Forecasts . . . . . . . . 87
3.2.1 The Ratio-Based Test . . . . . . . . . . . . . . . 87
3.2.2 The Difference-Based Test . . . . . . . . . . . . . 89
3.2.3 Bayesian Approach . . . . . . . . . . . . . . . . . 91
3.3 A Small Scale Monte Carlo Study . . . . . . . . . . . 94
3.3.1 Simulation Design . . . . . . . . . . . . . . . . . 94
3.3.2 Simulation Results . . . . . . . . . . . . . . . . 95
3.4 Empirical Application . . . . . . . . . . . . . . . . 96
3.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.2 Ratio-Based and Difference-Based Tests . . . . . . 97
3.4.3 Bayesian Methods . . . . . . . . . . . . . . . . . 98
3.5 Extension for Tests Under Regime-Switching . . . . . 101
3.5.1 Testing Equation . . . . . . . . . . . . . . . . . 101
3.5.2 The Estimation Algorithm . . . . . . . . . . . . . 103
3.5.3 Simulation Example . . . . . . . . . . . . . . . . 106
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . 107
References . . . . . . . . . . . . . . . . . . . . . . . 109

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