過去研究隨機邊界模型於經濟相關議題的文獻，多數直接假設誤差項分配，若做檢定，也只針對模型組合誤差 (composed error) 中的其中一項做檢定，而未對整個模型的組合誤差分配做檢定。然而，利用最大概似法估計隨機邊界模型時，誤差項的分配的錯誤設定可能會導致不一致的估計式，故在估計隨機邊界模型前確認誤差項的設定有其必要性。過去雖然有學者提出幾種不同檢定誤差項設定的方法，例如 LM 檢定、偏態檢定、LR檢定與動差基礎檢定，但這些方法都有許多的限制，所以後續學者針對過去的不足提出更具有彈性、能針對整個組合誤差做檢定的方法—sine/cosine 檢定，故本研究欲針對可取得實證資料且應用隨機邊界模型的研究進行 cosine 檢定確認模型誤差項的分配設定是否正確。本研究欲對模型假設為常態-半常態分配、常態-指數分配或常態-截斷常態分配之下進行檢定，但既有的檢定套件僅提供對前兩者檢定的運算，故本研究將針對常態-截斷常態分配導出cosine 檢定統計量，以進行常態-截斷常態分配的隨機邊界模型誤差項分配設定的檢定。最後，研究結果顯示過去8個隨機邊界模型設定中，其中5個模型誤差項分配設定沒有問題，有3個模型的誤差項分配設定可能有誤。

In the past, many studies in the stochastic frontier literature only assumed the distribution of the composed error but did not test the distribution assumption. In some cases where tests were conducted, the tests are most likely on one of the terms in the composed error, instead of the entire composed error itself. However, when we estimate the stochastic frontier model by the maximum likelihood method, a wrong specification of the composed error will lead to inconsistency in the maximum likelihood estimators. 
Some tests have been proposed in the literature, including the LM test, skewness test, LR test and the moment-based specification test, but they are restricted to the test of one of the terms in the composed error. Therefore, Chen and Wang (2012) introduced the sine/cosine test which is more flexible and comprehensive in testing the distribution assumptions of the composed error term. In our research, we use the cosine test to evaluate results of published papers where stochastic frontier models are used in the analysis. Because Chen and Wang (2012) only provided the package of testing half-normal or exponential assumption, we derive the test statistic of truncated-normal by the idea in their paper. The results show that 3 out of 8 empirical papers may have problems in the misspecification of the composed error distribution.

目錄
第一章 緒論	1
第二章 文獻回顧	4
第一節 隨機邊界模型	4
第二節 最大概似估計法	5
第三節 隨機邊界模型誤差項分配設定的檢定	7
第三章 研究方法	12
第一節 中央殘差基礎動差估計	12
第二節 截斷常態分配假設下的 cosine 檢定	15
第三節 研究方法	19
第四章 資料分析與討論	20
第五章 結論與建議	38
參考文獻	40

表目錄
表1 應用隨機邊界模型於經濟議題的文獻	20
表2 偏態檢定、LR 檢定與 cosine 檢定結果 (Greene, 1990)	22
表3 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Greene, 1990)	22
表4 偏態檢定、LR 檢定與 cosine 檢定結果 (Coelli et al., 2005)	24
表5 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Coelli et al., 2005)	24
表6 偏態檢定、LR 檢定與 cosine 檢定結果 (Farsi et al., 2005)	26
表7 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Farsi et al., 2005)	26
表8 偏態檢定、LR 檢定與 cosine 檢定結果一 (Greene, 2005)	28
表9 偏態檢定、LR 檢定與 cosine 檢定結果彙整一 (Greene, 2005)	28
表10 偏態檢定、LR 檢定與 cosine 檢定結果二 (Greene, 2005)	30
表11 偏態檢定、LR 檢定與 cosine 檢定結果彙整二 (Greene, 2005)	30
表12 偏態檢定、LR 檢定與 cosine 檢定結果 (Greene, 2007)	32
表13 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Greene, 2007)	32
表14 偏態檢定、LR 檢定與 cosine 檢定結果 (Belotti et al., 2012)	34
表15 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Belotti et al., 2012)	34
表16 偏態檢定、LR 檢定與 cosine 檢定結果 (Chaudhuri et al., 2015)	36
表17 偏態檢定、LR 檢定與 cosine 檢定結果彙整 (Chaudhuri et al., 2015)	36

參考文獻
1.	Aigner, D., Lovell, C. K., Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of econometrics, 6(1), 21-37.
2.	Belotti, F., Daidone, S., Ilardi, G., & Atella, V. (2013). Stochastic frontier analysis using Stata. The Stata Journal, 13(4), 719-758.
3.	Chaudhuri, K., Chowdhury, P., Kumbhakar, S. C. (2015). Crime in India: specification and estimation of violent crime index. Journal of Productivity Analysis, 43(1), 13-28.
4.	Chen, Y. T., Wang, H. J. (2012). Centered-residuals-based moment estimator and test for stochastic frontier models. Econometric Reviews, 31(6), 625-653.
5.	Coelli, T. (1995). Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis. Journal of productivity analysis, 6(3), 247-268.
6.	Coelli, T. J., Rao, D. S. P., O'Donnell, C. J. and Battese, G. E. (2005). An introduction to efficiency and productivity analysis. 2nd ed. New York: Springer.
7.	Farsi, M., Filippini, M., & Greene, W. (2005). Efficiency measurement in network industries: application to the Swiss railway companies. Journal of Regulatory Economics, 28(1), 69-90.
8.	Greene, W. H. (1990). A gamma-distributed stochastic frontier model. Journal of econometrics, 46(1-2), 141-163.
9.	Greene, W. (2005). Fixed and random effects in stochastic frontier models. Journal of productivity analysis, 23(1), 7-32.
10.	Greene, W. H. (2007). Econometric modeling guide. LIMDEP version, 10.
11.	Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica: Journal of the Econometric Society, 1029-1054.
12.	Kopp, R. J., Mullahy, J. (1990). Moment-based estimation and testing of stochastic frontier models. Journal of Econometrics, 46(1-2), 165-183.
13.	Lee, L. F. (1983). A test for distributional assumptions for the stochastic frontier functions. Journal of Econometrics, 22(3), 245-267.
14.	Meeusen, W., van Den Broeck, J. (1977). Efficiency estimation from Cobb-Douglas production functions with composed error. International economic review, 435-444.
15.	Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica: Journal of the Econometric Society, 53, 1047-1070.
16.	Schmidt, P. (1985). Frontier production functions. Econometric reviews, 4(2), 289-328.
17.	Schmidt, P., Lin, T. F. (1984). Simple tests of alternative specifications in stochastic frontier models. Journal of Econometrics, 24(3), 349-361.
18.	Stevenson, R. E. (1980). Likelihood functions for generalized stochastic frontier estimation. Journal of econometrics, 13(1), 57-66.
19.	Tauchen, G. (1985). Diagnostic testing and evaluation of maximum likelihood models. Journal of Econometrics, 30(1-2), 415-443.
20.	Wang, W. S., Amsler, C., & Schmidt, P. (2011). Goodness of fit tests in stochastic frontier models. Journal of Productivity Analysis, 35(2), 95-118.