條件動差限制式的新估計與檢定方法

在經濟和計量模型中，條件動差限制式(conditional moment restrictions)是相當常見的。如何藉由條件動差限制式來估計模型中的參數也因此是一個重要的議題。在文獻上，一般較典型的作法都是先從中選取有限個數的無條件動差限制式 (unconditional moment restrictions)後，再加以分析。然而，這樣的方法很可能無法提供足夠的資訊來認定模型中的參數。其主要的原因在於根據條件動差限制式而引伸出的無條件動差限制式實際上具有無窮條，而我們在分析時卻只考慮了其中的有限條。因此，條件動差限制式中的參數認定 (identification) 問題，實際上取決於如何考慮所有的無條件動差限制式。在第一章中，我們提出一個條件動差限制式的一致性 (consistent) 參數估計方法。這個新方法主要是利用generically comprehensive revealing 函數有系統地將所有無條件動差限制式納入分析。 Stinchcombe and White (1998) 已經說明這樣類型的函數在理論上是優於Dominguez and Lobato (2004)的方法中所使用的指標函數 (indicator function)。除此之外，為了得到一個容易計算的一致性估計方法，我們更進一步運用傅利葉分析法將這些無條件動差限制式作巧妙的結合，其所建議求解的目標函數即為此傅利葉係數的平方和。這樣的轉換有下列兩點好處：第一，每一個傅利葉係數都可視為所有無條件動差限制式的一種加權平均。第二，較後面項數的傅利葉係數因為很接近零而對於參數的認定提供甚少的資訊，因此在實際運用上我們可以僅考慮前面幾項的傅利葉係數而仍保有能認定參數的足夠資訊。由於這兩個優點，使我們的估計方法能在考慮所有無條件動差限制式下仍能保有相對簡單的形式。除此之外，在所有 generically comprehensive revealing 函數中，我們更進一步建議選用指數函數 (exponential function)來分析，其原因在於所得到的傅利葉係數將具有可分析的形式(analytic form)。這將會使我們的方法在實際運用上更為便利。我們證明這樣的估計式具有一致性及極限常態 (asymptotic normality)的特性，並且模擬結果也顯示我們所提出的估計式有相當良好的性質。對於任何條件動差限制式下的一致性估計式而言，Chamberlain(1987) 提出了所謂的效率性界限 (efficiency bound) 的概念，此界限即為估計式的變異數在極限上所可能達到最小值。當一個估計式的變異數在極限上能達到此界限，我們即稱此估計式具有效率性。由於第一章所提出的一致性估計式並不具有效率性，因此我們在第二章中提出一個容易運算且具有一致性和效率性的估計方法。我們首先延伸了第一章的作法，將傅利葉係數拆解成實部 (real part) 和虛部 (imaginary part) 兩部分，再利用實證概似(empirical likelihood) 法來估計。利用 Donald et al. (2003) 所提出對應於無窮多條限制式下的極限理論，我們證明了這樣的估計式具有一致性及效率性。此外，藉由模擬的結果，估計式的效率性也清楚反映在較小的標準誤 (standard error) 上。在第三章中，我們則致力於提出一個關於條件均數函數(conditional mean function) 的一致性線性檢定方法。線性模型在計量分析上可視為刻畫變數間關係的基準模型，也因此檢定模型是否為線性是一個相當重要的議題。由於線性模型檢定的虛無假設 (null hypothesis) 可以被表示成條件動差限制式，因此我們可以直接利用前兩章所提出的分析方法。據此，我們所建議的檢定方法是建立在衡量傅利葉係數的平方和大小上。一旦平方和遠大於零，我們就傾向拒絕模型為線性的假設。在這一章中，我們不但建立了這個檢定方法的極限理論，並且利用了 wild bootstrap 方法來進行推論。模擬的結果顯示了我們所提出了檢並方法優於其他文獻上常用的 Cramer-von Mises 和 Kolmogorov-Smirnov 檢定。

關鍵字

條件動差限制式；效率性界限；實證概似法；傅利葉級數；傅利葉係數；線性檢定；參數認定

並列摘要

Conditional moment restrictions are common in economic and econometric models. Based on conditional moment restrictions, how to estimate the parameters of interest is thus an important task. In the literature, it is typical to start by considering a finite set of unconditional moment restrictions implied by the conditional restrictions. This approach, however, may not provide enough information for identifying the parameter of interest. This is because that, to be equivalent to the conditional moment restrictions, there must be infinitely many unconditional restrictions. The parameter identifiability thus hinges on whether all these unconditional restrictions can be considered. In Chapter 1, we propose a new estimation method that yields a consistent estimator for conditional moment restrictions models. The approach is based on a continuum of unconditional moment restrictions which is generated by a set of generically comprehensive revealing functions. In view of Stinchcombe and White (1998), this class of functions is theoretically preferred to the indicator functions of Dom´ınguez and Lobato (2004). To construct a consistent estimator, we further delicately combine this continuum of unconditional moment restrictions by projecting them along the exponential Fourier series. The resulting objective function is thus the sum of squares of the corresponding Fourier coefficients. Two attractive features of this transformation immediately follow. First, each transformed moment restriction can be viewed as a weighted average of all continuum of moment restrictions. Second, the “remote” Fourier coefficients tend to zero and hence provide little information about the parameters of interest. This indicates that only the more important Fourier coefficients are needed for estimation. As a consequence, the proposed objective function is relatively simple while still involving the full continuum of moment conditions. Furthermore, among all generically comprehensive revealing functions, we show that the unconditional moments and hence the Fourier coefficients have analytic forms when the exponential function is used. This greatly facilitates parameter estimation in practice. The proposed estimator is shown to be consistent and asymptotically normal. Monte Carlo simulations also show that it has good finite sample performance. Given the conditional moment restrictions, Chamberlain (1987) provided the efficiency bound for the asymptotic variance-covariance matrix of any consistent estimator. Since the extreme estimator proposed in Chapter 1 is not efficient, we further extend the approach by decomposing Fourier coefficients into the real and imaginary parts, and then employ the empirical likelihood method in Chapter 2. Under the framework of many moment restrictions, the asymptotics of the proposed estimator is established based on the results of Donald et al. (2003). The proposed estimator is consistent, and its asymptotic variance-covariance attains Chamberlain’s efficiency bound. Our Monte Carlo simulations also demonstrate that the efficiency gain from the proposed estimator is rather significant. In Chapter 3, we focus on constructing a consistent linearity test in conditional mean function. In econometric modelling, the linear model is a leading one in applications because it is usually treated as a reference (benchmark) model in characterizing economic relationships. Since the null hypothesis can be represented as a set of conditional moment restrictions, the approach in first two chapters can be employed straightforwardly. As a consequence, we suggest to test linearity based on the summation of the squared magnitudes of the corresponding Fourier coefficients. Once the distance between this summation is very different from zero, we reject the null. The asymptotics of the proposed test under the null and alternative hypotheses are well established, and the wild bootstrap procedure is introduced for statistical inference. Monte Carlo simulations show that the proposed test compares favorably with the Cramer-von Mises and Kolmogorov-Smirnov type test statistics in hypothesis testing. Keywords: conditional moment restrictions, efficiency bound, empirical likelihood, Fourier series, Fourier coefficients, generically comprehensive revealing functions, linearity test, parameter identification, wild bootstrap.

並列關鍵字

conditional moment restrictions ； efficiency bound ； empirical likelihood ； Fourier series ； Fourier coefficients ； generically comprehensive revealing functions ； linearity test ； parameter identification ； wild bootstrap

參考文獻

References-- Chapter 1

Chen, X. and H. White (1998). Central limit and functional central limit theorems for

Stuart, R. D. (1961). An Introduction to Fourier Analysis, New York: Halsted Press.

References-- Chapter 2

Ai, C. and X. Chen (2003). Efficient estimation of models with conditional moment restrictionscontaining unknown functions, Econometrica, 71, 1975–1843.

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條件動差限制式的新估計與檢定方法

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