本論文主要在探討隨機邊界法運用於追蹤資料上的模型設定以及可能會遇到的估計問題。隨機邊界法廣泛運用於估計銀行，公司，政府以及醫療體系等的營運績效。近年來由於資料的整理與建構愈趨完善，追蹤資料更是被頻繁地使用；而當隨機邊界法運用在追蹤資料上時，除了可以提供樣本增加的好處之外，其衍伸的問題如固定效果以及在效率設定的假設皆被提出且被廣泛的探討。然而近年來在一般迴歸中所關心的橫斷面相依性則鮮少有隨機邊界的文章作進一步的討論。 計量學家採用因素結構嘗試去刻畫此一橫斷面相依性，而其因素結構可以被分為引起經濟體系變化的共同衝擊與個體對於衝擊所產生個體獨有的效果。 相較於既存沒有探討抑或者是將因素結構視為效率的文獻，本文所考慮的隨機邊界模型則將因素結構(與固定效果)視為無法觀察的變異而非效率，並進一步說明其必要性，據此提出一模型轉化的方式用來消除因素結構以得到一致性的估計。本論文首先介紹隨機邊界模型的基本概念與近年來運用在追蹤資料時所關心的估計議題，爾後將兩篇文章分別編列為第二章與第三章，並提供摘要如下：第二章運用因素結構建立了一隨機邊界模型，此模型的優勢在於可以將效率與無法觀察到的共同衝擊所導致的影響分開。本文運用了與Pesaran~(2006)相同的概念來消除共同相關效果，於此同時可以利用最大概似估計法得到具有一致性的估計，且證明在$N$和$T$都趨近到無限大，且$T/N$趨近到0時，這樣的轉換會得到與無共同效果但經過相同轉換一樣的極限分配。本文也利用模擬來佐證此方法在小樣本的表現。最後我們將此方法運用在美國儲蓄與商業銀行的成本效率分析，實證結果支持儲蓄銀行相較於商業銀行是較無效率的。但不管是儲蓄銀行還是商業銀行在2008年金融危機前，整體效率是呈現穩定上升的。在第三章中，我們將效率的設定由外生變數決定改為隨機性的，並且延續上一章的設定容許共同相關效果的存在。在此設定下則可以試圖捕捉個體不受外生變數所影響的效率。在本章中，為了能夠得到較為穩健的估計結果，我們將被轉換的模型運用EM演算法來估計。在模擬中，我們的方法相較於過去忽略掉共同相關效果的估計方法有著相對於精確的表現，也佐證了忽略掉共同相關效果對於估計上會存在相當程度的偏誤。 This dissertation consists of two independent yet related essays which study the topic of the stochastic frontier model. The stochastic frontier model is widely used in assessing the efficiency or performance in banking, firms, the government and medical systems. Recently, cross-sectional data has been supplanted by the use of panel data that has the advantage of a large sample size. In addition, the accompanying econometric issues including fixed effects and the specification of efficiency have been introduced and discussed at length. Despite these issues, many studies have started to use the factor structure to describe the heterogeneous impacts caused by common shocks, which model involves this factor structure also accommodating the property of cross-sectional dependence. However, there is still less discussion in the literature on stochastic frontier analysis. Compared with the existing literature, we treat the factor structure and efficiency as separate components instead of treating all factor structure as efficiency. We further discuss the reasons why the factor structure and efficiency should be separated. A transformation has been proposed in this dissertation to obtain a consistent estimate via the maximum likelihood based approach.The first chapter of this dissertation introduces the main issues in the stochastic frontier model in regard to the panel data. The two essays are organized as chapter 2 and chapter 3. I provide brief summaries of these two chapters as follows.Chapter 2 develops a panel stochastic frontier model with unobserved common shocks to capture cross-sectional dependence among individual firms. The novel feature of our model is to separate technical inefficiency from the effects induced by unobserved common shocks and individual heterogeneity. We propose a modified maximum likelihood method that does not require estimating unobserved common correlated effects and discuss the asymptotic properties of the proposed estimation procedure. The basic idea of our approach is similar to that in Pesaran (2006) for the linear panel regression. We show that the proposed method can control the common correlated effects and obtain consistent estimates of parameters for the panel stochastic frontier model. Our Monte Carlo simulations show that the modified MLE has satisfactory finite sample properties under a significant degree of cross-section dependence for relatively small T. The proposed method is also illustrated in applications based on a comparison of the efficiency of savings and the commercial banking industry in the US.In chapter 3, we consider a linear model with time-invariant fixed effects to represent heterogeneity and the cross-sectional dependence by introducing common correlated effects, and the time-variant technical inefficiency and idiosyncratic errors jointly characterized by a multivariate skew normal distribution. To consistently estimate the slope coefficients and variances in the above model, we propose a transformation which is similar to that introduced in chapter 2 to eliminate fixed effects and common correlated effects. Based on the transformed likelihood function, we then introduce an EM Algorithm to robustly estimate these parameters. Our Monte Carlo simulation shows that the proposed method is quite accurate in the presence of common correlated effects, while conventional models that do not take these effects into account can result in severely biased parameter estimates.
