ESSAYS IN INDUSTRIAL ORGANIZATION By Jaemin Ryu A DISSERTATION Submitted to Michigan State University in partial fulllment of the requirements for the degree of Economics  Doctor of Philosophy 2021 ABSTRACT ESSAYS IN INDUSTRIAL ORGANIZATION By Jaemin Ryu Chapter 1. Measurement and Decomposition of Cost Ineciency Using Copulas: An Application to the U.S. Banking Industry This paper proposes a model and an estimation strategy using copulas in order to measure and decompose technical and allocative ineciency in the translog cost system. This study adapts the stochastic cost frontier model from Kumbhakar (1997) and employs the APS copulas developed by Amsler et al. (2021) to capture the dependence between technical and allocative ineciency. The joint density of the system is derived by the probability integral transform and the copula-based version of the Rosenblatt transformation, leading to the method of simulated likelihood estimation. This study also proposes a strategy to estimate individual ineciency using density deconvolution and conditional distributions. The new methods are then applied to the U.S. banking industry. The results suggest that U.S. bank costs increased by approximately 20% in 2019 and 2020 due to ineciency, where technical and allocative ineciency represented 16∼18% and 2.5%, respectively. In addition, ignoring the dependence between technical and allocative ineciency would produce less plausible results. Chapter 2. Measurement and Decomposition of Cost Ineciency Using Copulas: Evidence from Monte Carlo Simulations The purpose of this paper is to provide methods for copula-based simulations and to demon- strate the performance of the estimation strategy that can measure and decompose cost ineciency. First, a method to draw random numbers using the APS-3-A copula, which cor- responds to the three-input case, is presented. Specically, copula arguments can be obtained from random numbers distributed independently and uniformly over [0, 1] by applying the inverse Rosenblatt transformation, which needs to derive conditional distributions of cop- ulas. Then, dependent random numbers can be generated by the inverse transformation method. Second, quasi-Monte Carlo simulations are conducted given the data generating process. Simulation results suggest that the parameters of the translog cost system that accommodates technical and allocative ineciency can be reliably estimated when the APS copulas are employed. It would also yield biased estimates when the disturbances in the cost function and the cost share equations of the system are regarded as independent. Chapter 3. Demand Estimation of Deposits: A Case of the Korean Financial Industry This paper estimates a structural demand model for deposits in the Korean nancial sector in order to measure the eects of deregulation in payment and settlement systems in 2009, which caused cash management accounts (CMAs) of securities companies to become close substitutes for traditional deposit services provided by banks. Following the discrete choice literature, depositors choose among dierentiated nancial institutions, considering their of- fered interest rates and other attributes. Although it is also assumed that market discipline in banking exists, it depends on the nancial stability situation. The results show that con- sumers respond favorably to deposit rates, the branch stang, and the number of branches of depository institutions in tranquil times. On the other hand, they consider the nancial institution's capital adequacy ratio more important than interest rates during the nancial turmoil. This is similar to the phenomenon referred to as the ight to quality in other nan- cial markets. Therefore, although CMAs have the benet of higher interest rates compared to traditional deposit services, their market share has remained at low levels due to the pro- longed nancial stress since the global nancial crisis, which results in marginal increases in consumer welfare from the deregulation. This implies that the deregulation would not have successfully achieved the purpose of improving consumer welfare by promoting competition. ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor Kyoo il Kim for his generous support, encouragement, and valuable comments. I would also like to appreciate Peter Schmidt for his deep insights and clear guidance. I additionally thank Jay Pil Choi and Sriram Narayanan for their assistance and helpful comments. Last but not least, I deeply indebted my family and Hyunjung Kim for all their devoted support and love throughout my studies. I would like to acknowledge nancial support from the Bank of Korea and the Michigan State University Department of Economics. iv TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ix CHAPTER 1 MEASUREMENT AND DECOMPOSITION OF COST INEFFICIENCY USING COPULAS: AN APPLICATION TO THE U.S. BANKING INDUSTRY . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Translog Cost System of Kumbhakar (1997) Revisited . . . . . . . . . 5 1.2.2 Modied Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Estimation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Relationship between uTi and ξ i : APS Copulas . . . . . . . . . . . . . 15 1.3.2 Derivation of the Joint Density . . . . . . . . . . . . . . . . . . . . . 18 1.3.3 Maximum Simulated Likelihood Estimator . . . . . . . . . . . . . . . 24 1.4 Firm-level Ineciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Application: U.S. Banking Industry . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.2 Average Ineciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.3 Firm-level Ineciency . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 APPENDICES . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 38 APPENDIX A: Additional Details . . . . . . . . . . . . . . . . . . . . . . . . . . 39 APPENDIX B: Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . 46 CHAPTER 2 MEASUREMENT AND DECOMPOSITION OF COST INEFFICIENCY USING COPULAS: EVIDENCE FROM MONTE CARLO SIMULATIONS . . . . . . . . 49 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Simulating from the APS-3-A Copula . . . . . . . . . . . . . . . . . . . . . . 52 2.2.1 Derivation of Conditional Distributions . . . . . . . . . . . . . . . . . 52 2.2.2 Obtain Copula Arguments . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Generating Pseudo Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4 Results of Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . 62 2.4.1 Simulation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4.2 Simulation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 APPENDICES . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 68 APPENDIX A: Simulating from the APS-2-A Copula . . . . . . . . . . . . . . . . 69 APPENDIX B: Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . 71 v CHAPTER 3 DEMAND ESTIMATION OF DEPOSITS: A CASE OF THE KOREAN FINANCIAL INDUSTRY . . . . . . . . 75 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Empirical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3 Data and Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.2 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.1 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.2 Consumer Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 APPENDICES . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 APPENDIX A: An Overview on the Korean Financial System . . . . . . . . . . . 94 APPENDIX B: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 100 vi LIST OF TABLES Table 1.5.1: Interest Expenses and Other Earning Assets of the U.S. Banks . . . . . 28 Table 1.5.2: Estimation Results for 2019 . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 1.5.3: Estimation Results for 2020 . . . . . . . . . . . . . . . . . . . . . . . . . 30 Table 1.5.4: Average Ineciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Table 1.5.5: Standard Deviations of ξˆi2 and ξˆi3 . . . . . . . . . . . . . . . . . . . . . 33 Table 1.5.6: Average of ûA i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Table 1.5.7: Average of ûTi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Table 1.5.8: Average of ûTi and ûA i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Table 1.B.1: Descriptive Statistics of Key Variables for 2019 . . . . . . . . . . . . . . 46 Table 1.B.2: Descriptive Statistics of Key Variables for 2020 . . . . . . . . . . . . . . 46 Table 1.B.3: Classication of Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Table 2.4.1: Result of Simulation I (J = 2, M = 1, θ12 = 0.4) . . . . . . . . . . . . . 63 Table 2.4.2: Result of Simulation I (J = 2, M = 2, θ12 = 0.4) . . . . . . . . . . . . . 63 Table 2.4.3: Result of Simulation I (J = 3, M = 1, θ12 = θ13 = 0.2) . . . . . . . . . . 64 Table 2.4.4: Result of Simulation I (J = 3, M = 2, θ12 = θ13 = 0.2) . . . . . . . . . . 64 Table 2.4.5: Result of Simulation II (J = 2, M = 1, θ12 = 0.4) . . . . . . . . . . . . 66 Table 2.4.6: Result of Simulation II (J = 2, M = 2, θ12 = 0.4) . . . . . . . . . . . . 66 Table 2.4.7: Result of Simulation II (J = 3, M = 1, θ12 = θ13 = 0.2) . . . . . . . . . 66 Table 2.4.8: Result of Simulation II (J = 3, M = 2, θ12 = θ13 = 0.2) . . . . . . . . . 67 Table 2.B.1: Result of Simulation I (J = 2, M = 1, θ12 = 0) . . . . . . . . . . . . . . 72 Table 2.B.2: Result of Simulation I (J = 2, M = 1, θ12 = 0.2) . . . . . . . . . . . . . 72 vii Table 2.B.3: Result of Simulation I (J = 2, M = 2, θ12 = 0) . . . . . . . . . . . . . . 73 Table 2.B.4: Result of Simulation I (J = 2, M = 2, θ12 = 0.2) . . . . . . . . . . . . . 73 Table 2.B.5: Result of Simulation I (J = 3, M = 1, θ12 = θ13 = 0) . . . . . . . . . . . 74 Table 2.B.6: Result of Simulation I (J = 3, M = 2, θ12 = θ13 = 0) . . . . . . . . . . . 74 Table 3.2.1: Households' Preferences for Financial Instruments . . . . . . . . . . . . 82 Table 3.A.1: Total Assets of Major Financial Institutions in Korea . . . . . . . . . . . 94 Table 3.A.2: List of Banks in Korea (as of Q4 2016) . . . . . . . . . . . . . . . . . . . 95 Table 3.A.3: Major Financial Events in Korea . . . . . . . . . . . . . . . . . . . . . . 97 Table 3.B.4: Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Table 3.B.5: Classication of Financial Institutions . . . . . . . . . . . . . . . . . . . 98 Table 3.B.6: Distribution of Own Price Elasticities . . . . . . . . . . . . . . . . . . . 98 Table 3.B.7: Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 viii LIST OF FIGURES Figure 1.2.1: Degree of Technical and Allocative Ineciency . . . . . . . . . . . . . . 8 Figure 1.2.2: Relationship between ξ2 and ηi2 . . . . . . . . . . . . . . . . . . . . . . 9 Figure 1.3.1: PDF and CDF of ξij . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 1.3.2: Sample Correlations (The APS-2-A Copula) . . . . . . . . . . . . . . . 17 Figure 1.3.3: Procedure for a Change of Variables . . . . . . . . . . . . . . . . . . . . 20 Figure 1.4.1: Process to Measure and Decompose Individual Ineciency . . . . . . . 25 Figure 1.5.1: fˆe2 (e2 ) and fˆη2 |e2 (η2 |e2 ) for 2019 . . . . . . . . . . . . . . . . . . . . . . 32 Figure 1.5.2: fˆe3 (e3 ) and fˆη3 |e3 (η3 |e3 ) for 2019 . . . . . . . . . . . . . . . . . . . . . . 32 Figure 1.5.3: Distribution of ûAi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Figure 1.5.4: Distribution of ûTi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 1.B.1: Sample Correlations (The APS-3-A Copula) . . . . . . . . . . . . . . . 47 Figure 1.B.2: fˆe2 (e2 ) and fˆη2 |e2 (η2 |e2 ) for 2020 . . . . . . . . . . . . . . . . . . . . . . 48 Figure 1.B.3: fˆe3 (e3 ) and fˆη3 |e3 (η3 |e3 ) for 2020 . . . . . . . . . . . . . . . . . . . . . . 48 Figure 2.3.1: Procedure to Simulate Z = (uTi , ξi2 , · · · , ξiJ ) . . . . . . . . . . . . . . . 59 Figure 2.B.1: Sample Correlations between ζ1 and ζ2 . . . . . . . . . . . . . . . . . . 71 Figure 3.1.1: Interest Rates and Total Amount of CMAs . . . . . . . . . . . . . . . . 78 Figure 3.4.1: BIS ratios of Korean Banks . . . . . . . . . . . . . . . . . . . . . . . . 90 Figure 3.A.1: Financial Stability Indices of Korea . . . . . . . . . . . . . . . . . . . . 97 ix CHAPTER 1 MEASUREMENT AND DECOMPOSITION OF COST INEFFICIENCY USING COPULAS: AN APPLICATION TO THE U.S. BANKING INDUSTRY 1.1 Introduction Stochastic frontier models (SFMs) developed by Aigner et al. (1977) and Meeusen and van Den Broeck (1977) have been widely used for eciency analysis. There are two approaches to measure eciency in the SFMs. The rst one is an output-oriented approach, which is used to estimate the production frontier and to measure technical (in)eciency. The second one is an input-oriented approach, which can be used to estimate the cost frontier and to measure cost (in)eciency. As Kumbhakar and Lovell (2000) note, there are several dierences between these two approaches, an important dierence of which is that cost eciency can be decomposed into input-oriented technical eciency and input allocative 1 eciency, whereas output-oriented technical eciency cannot be decomposed. Farrell (1957) denes technical and allocative ineciency as follows. Technical ineciency occurs when a producer fails to produce the maximum output from a given input bundle. Allocative ineciency occurs when a producer uses inputs in the wrong proportions, given input prices. As ineciency can arise by these dierent causes, it is important to measure and decompose cost ineciency in order to evaluate the performance of rms. Schmidt and Lovell (1979) show how to measure both technical and allocative ineciency, assuming the Cobb-Douglas production technology. Nevertheless, it would be necessary to 2 apply exible functional forms, such as a translog function , when measuring cost ineciency. Since Christensen et al. (1971) and Christensen and Greene (1976), the translog functional form has played an important role in cost studies, owing to several virtues that overcome the 1 Hereafter, technical (in)eciency means input-oriented technical (in)eciency. 2 For more details about the translog function, please refer to Kumbhakar and Lovell (2000) and Sickles and Zelenyuk (2019). 1 limitations of the Cobb-Douglas function. As mentioned in Kumbhakar and Lovell (2000), for example, the translog cost function can accommodate multiple outputs without violating the requisite curvature conditions in output space, while the Cobb-Douglas functional form cannot. In addition, the translog function can provide a second-order approximation to any well-behaved underlying cost frontier, whereas the Cobb-Douglas representation would lead to biased estimates of ineciency; this is because unmodeled technology complexity could appear in the error term, which contains information about ineciency, due to its simplicity. 3 In contrast, as rst noted by Greene (1980) , econometric issues arise when employing the translog functional form. The problem is characterized as follows. Given the denition in Farrell (1957), the cost system that allows for technical and allocative ineciency can be written as lnCi = lnC(y i , wi ) + ϵi = lnC(y i , wi ) + vi + ui = lnC(y i , wi ) + vi + uTi + uA i sij = sj (y i , wi ) + eij , j = 2, · · · , J, (1.1.1) where Ci is the actual cost of producer i, C(y i , wi ) is the deterministic kernel of the stochastic cost frontier, y i ∈ RM + is a vector of M outputs produced by producer i, wi ∈ RJ++ is a vector of input prices faced by producer i, vi ∈ R is a random disturbance, uTi ∈ R+ represents a cost increase due to technical ineciency, uA i ∈ R+ represents a cost increase due to allocative ineciency, ui = uTi + uA i , ϵi = vi + ui , sij ∈ [0, 1] is the producer i's actual cost share of input j , sj (y i , wi ) ∈ [0, 1] is the optimum cost share of input j derived from Shephard's lemma, and eij ∈ R is the disturbance due to allocative ineciency of producer i and noise. The important question is how to model ui in the cost function and eij in the cost share equations. As long as eij represents allocative ineciency, it cannot be independently distributed of ui that captures both technical and allocative ineciency. If it is assumed 3 It is known as the Greene Problem. Please refer to Bauer (1990) and Kumbhakar and Lovell (2000) for detailed discussion. 2 that ui and eij are independent, it would lead to inconsistent parameter estimates, and it is impossible to decompose cost ineciency into two sources (Kumbhakar and Lovell, 2000, p.156). So far, three types of solutions to the Greene Problem have been proposed. The rst solution is nding the analytic relationship between uAi and ei = (ei2 , · · · , eiJ ) (Schmidt and Lovell (1979) and Kumbhakar (1989, 1997)). The second solution is approximating uA i as a function of ei , such as in Schmidt (1984), Mel (1984), Bauer (1985), and Kumbhakar (1991). The third solution proposed by Greene (1980) is ignoring the relation between them and assuming that ϵi and ei are independent. However, as discussed in Bauer (1990), all of these existing solutions are not ideal. The rst one can be used when restrictive functional forms, such as a Cobb-Douglas functional form, and/or restrictive assumptions are imposed. The second solution is valid only when the approximation function captures the true relationship between uAi and ei . The third approach might fail to use all available information for estimation. The main contribution of this paper is to propose a translog cost system, which is rig- orously developed based on economic theory, providing a solution to the Greene Problem that overcomes the limitations of previous studies. In other words, technical and allocative ineciency can be precisely measured and decomposed by using the proposed model with a exible functional form. In addition to the main contribution, a novel estimation strategy is proposed that is consistent with the theory behind the stochastic cost frontier model as well as computationally easy. Lastly, this study suggests a method to estimate individual ineciency when additive noise terms are allowed in the cost share equations. A stochastic cost frontier model taking a translog cost functional form is constructed based on Kumbhakar (1997) in order to derive the exact relationship between the error terms representing allocative ineciency in the cost function and the cost share equations. However, as in Schmidt and Lovell (1980), dependence between technical and allocative ineciency is assumed. This is modeled by the APS copulas developed by Amsler et al. 3 (2021). Several assumptions are imposed to capture the dependence and to make the model more realistic and estimable. The method of simulated likelihood is applied to estimate the model, where the joint density of the model is derived by the probability integral transform and the copula-based version of the Rosenblatt transformation. Additionally, given that the model can estimate only average ineciency of rms, this study also proposes a strategy to measure and decompose individual ineciency. Jondrow et al. (1982) and Battese and Coelli (1988) propose ways to estimate rm-level ineciency in production using the conditional distribution f (uTi |ϵi ), where uTi is output-oriented technical ineciency, ϵi = vi − uTi , and vi is a random disturbance. This method could be adapted to the stochastic cost frontier analysis. However, due to the dierent environment, we must employ density deconvolution to error terms in the cost share equations of (1.2.1). The new model and strategies are applied to U.S. depository institutions. Numerous 4 studies on bank eciency exist for various themes. For instance, innovations in technology, such as telecommunication technologies and information processing, have been intensively adopted in the banking industry (see Feng and Serletis, 2009). In addition, especially in the United States, regulatory changes, such as branching deregulation and enhanced regulatory capital requirements, have aected operation strategies of banks. Since these factors have an impact on technical and allocative ineciency of banks, respectively, it is necessary to measure and decompose cost ineciency of banks in order to identify each factor's eect on the performance of banks. Furthermore, it is possible that technical and allocative ine- ciency of banks has changed due to outbreak of COVID-19, since the pandemic has aected nancial markets and resource allocation of depository institutions. The remainder of the chapter is organized as follows. Section 1.2 develops the econometric model. Section 1.3 presents the estimation strategy. Section 1.4 develops the strategy to estimate rm-level ineciency. Section 1.5 presents empirical results applied to the U.S. banking industry. Section 1.6 concludes the chapter. 4 Please refer to Berger and Humphrey (1997) and Bhatia et al. (2018) for more details. 4 1.2 Model 1.2.1 Translog Cost System of Kumbhakar (1997) Revisited Kumbhakar (1997) establishes an exact relationship between the terms representing alloca- tive ineciency in the cost function and the cost share equations of the stochastic cost frontier model. This subsection summarizes its setup and results. 1.2.1.1 Setup T Let P (y i , xi e−ui ) = 0 be a production possibility frontier , where 5 xi ∈ RJ+ is a vector of J inputs used by producer i. Recall that y i ∈ RM + is a vector of M outputs, and uTi ∈ R+ represents technical ineciency. P (·) is assumed to be dierentiable. Then, the cost minimization problem of producer i who is only technically inecient can be written as T min w′i xi s.t. P (y i , xi e−ui ) = 0. xi Note that it yields the same solution to the following problem such that min∗ w′i x∗i s.t. P (y i , x∗i ) = 0, xi T where x∗i = xi e−ui . Its rst-order conditions are Pj (y i , x∗i ) wij ∗ = , j = 2, · · · , J, P1 (y i , xi ) wi1 where Pj (y i , x∗i ), j = 1, · · · , J, denotes the partial derivative of P (y i , x∗i ) with respect to x∗ij . Given this result, the rst-order conditions of producer i who is both technically and allocatively inecient can be written as Pj (y i , x∗i ) wij ξj ∗ = e , j = 2, · · · , J, P1 (y i , xi ) wi1 5 Kumbhakar (1997) uses a production function to derive the translog cost system, which means he considers a single-output case. However, as noted by Parmeter and Kumbhakar (2014), the derivation and the result of the translog cost system for a multiple-output case are similar to the single-output case. 5 where ξj ∈ R, j = 2, · · · , J, represents producers' allocative ineciency for the input pair (j, 1). Note that ξ1 = 0 by construction. If ξj = 0 for j = 2, · · · , J , the input pair (j, 1) is perfectly allocatively ecient. 1.2.1.2 Results The stochastic cost frontier model can be written as lnCi = lnC(y i , wi ) + vi + uTi + uA i sij = sj (y i , wi ) + ηij , j = 2, · · · , J, where ηij ∈ R is the deviation from the optimum cost share of input j due to only allocative ineciency of producer i, which does not contain noise. Recall that Ci is the actual cost of producer i, C(y i , wi ) is the deterministic kernel of the stochastic cost frontier, vi ∈ R is a random disturbance, uA i ∈ R+ represents a cost increase due to allocative ineciency, sij ∈ [0, 1] is the actual cost share of input j, and sj (y i , wi ) ∈ [0, 1] is the optimum cost share of input j. Assume that the deterministic kernel of the stochastic cost frontier takes a translog functional form. Then, deterministic components of the system, lnC(y i , w i ) and sj (y i , wi ), can be written as M M M X y 1 X X yy lnC(y i , w i ) = β0 + βm (lnyim ) + β (lnyim )(lnyin ) m=1 2 m=1 n=1 mn J J J X 1 X X ww + βjw (lnwij ) + β (lnwij )(lnwik ) j=1 2 j=1 k=1 jk X M X J yw + βmj (lnyim )(lnwij ) m=1 j=1 X J XM yw sj (y i , wi ) = βjw + ww βjk (lnwik ) + βmj (lnyim ), j = 2, · · · , J. k=1 m=1 wij xij wij ∂C(y i ,wi ) ∂ lnC(y i ,wi ) Note that sj (y i , wi ) = C(y i ,wi ) = C(y i ,wi ) ∂wij = ∂ lnwij . The second equality holds because of Shephard's lemma. Also, the terms representing allocative ineciency can be 6 written as J J X J J J X X 1 X X ww uA i = βjw ξj + ww βjk (lnwij )ξk + β ξj ξk j=1 j=1 k=1 2 j=1 k=1 jk XM X J X J yw + βmj (lnyim )ξj + ln (s∗ij /eξj ) m=1 j=1 j=1 PJ ∗ ξk ξj PJ ww sj (y i , wi )[1 − { k=1 (sik /e )}e ] + k=1 βjk ξk ηij = , j = 2, · · · , J, { Jk=1 (s∗ik /eξk )}eξj P PJ where s∗ij = sj (y i , wi ) + k=1 ww βjk ξk , j = 2, · · · , J, is the shadow cost share of input j for producer i who is assumed to be only allocatively inecient. To satisfy the properties of the cost function, restrictions on parameters are required yy yy ww ww PJ w such that βmn = βnm ∀m ̸= n and βjk = βkj ∀j ̸= k for symmetry, j=1 βj = 1, PJ ww PJ yw k=1 βjk = 0 ∀j , and j=1 βmj = 0 ∀m for linear homogeneity in wi . Because of these PJ PJ j=1 ηij = 0 ∀i are guaranteed. 6 restrictions, j=1 sj (y i , w i ) = 1 and 1.2.2 Modied Model As noted by Kumbhakar and Lovell (2000), Kumbhakar (1997)'s model treats allocative ineciency of the translog cost system in a theoretically and econometrically consistent manner and provides a solution to the Greene Problem. Nevertheless, there is still room for improvement. In Section 1.2.2.1, several limitations of Kumbhakar (1997) are discussed and assumptions of his model are modied. In Section 1.2.2.2, the modied model and distributional assumptions are presented. 1.2.2.1 Assumptions First, Kumbhakar (1997)'s model does not capture the relationship between the one-sided term representing technical ineciency, uTi , and the two-sided terms representing alloca- tive ineciency, ξj s. It implies that they can be assumed to be independent; this means 6 Please see to Appendix 1.A.1 for the proof. 7 technically ecient producers can be allocatively inecient and vice versa. Rather than imposing such an assumption, it would be more reasonable to assume that they are depen- dent, which means that technically ecient producers tend to be allocatively ecient, and technically inecient producers are likely to be allocatively inecient. However, as Figure 1.2.1 illustrates, the degree of allocative ineciency does not depend on the value of ξj itself but the size of its absolute value instead. Since the degree of technical ineciency, uTi , is non-negative, it is dicult to model the relation between uTi and ξj . To the best of my knowledge, only two studies, namely Schmidt and Lovell (1980) and Amsler et al. (2021), 7 have modeled the aforementioned relationship between technical and allocative ineciency. However, both studies consider the single-output cost system assuming Cobb-Douglas pro- duction technology, rather than the translog cost system (as in Kumbhakar (1997)) that is most widely used for empirical cost studies and that accommodates multiple-output cases. Figure 1.2.1: Degree of Technical and Allocative Ineciency T (a) Technical Ineciency (ui ) (b) Allocative Ineciency (ξj ) Note: In both gures, the x-axis is the value of uTi or ξj , and the y-axis is its density value. Second, Kumbhakar (1997)'s model imposes a restrictive assumption that the magnitudes of allocative ineciency, ξj s, are invariant across producers. This implies that it only con- siders systematic tendency for over- or under-utilization of any input relative to any other 7 Following the notations in this paper, they assume that uTi is positively correlated with ηi = (ηi2 , · · · , ηiJ ) since terms like ξj s in Kumbhakar (1997) are not introduced to their model. Given that there are two terms that can be interpreted as allocative ineciency in the cost share equations, ξj s and ηi , the latter part of this section examines which term will be linked to uTi . 8 inputs. Despite the assumption on ξj s, their impacts on cost, uAi , and on input shares, ηi, are dierent across producers, as they are inuenced by outputs, yim , and input prices, wij , by construction. As stated in Kumbhakar and Lovell (2000), the model becomes extremely dicult to estimate without this assumption. To be specic, if the magnitudes of allocative ineciency are assumed to be random, denoted by ξ i = (ξi2 , · · · , ξiJ ), it is hard to derive the distribution of ηi from the distribution of ξi . This is because, although they can be one-to-one in the narrow domain, ηi is a nonlinear function of ξi that is not globally invert- ible. For example, Figure 1.2.2 shows the relationship between ξ2 and ηi2 given parameter 8 values when a rm produces one output using two inputs. Therefore, the change of vari- ables theorem cannot be directly applied. However, this assumption needs to be relaxed to incorporate idiosyncratic deviations from the cost minimizing input ratios. In addition, relaxing this assumption enables researchers to rigorously model the relationship between the terms representing technical and allocative ineciency. Figure 1.2.2: Relationship between ξ2 and ηi2 (a) ξ2 ∈ [−1, 1] (b) ξ2 ∈ [−10, 10] Lastly, stochastic components are not included in the cost share equations. Kumbhakar (1997) analytically derives the optimum cost share, sj (y i , wi ), and ηi from the use of Shep- 9 hard's lemma, thereby ηi represents pure allocative ineciency stemmed from the opti- 8 Although Kumbhakar (1997) does not provide the result, uA and ηij can be simplied when no additive i error terms in the cost share equations is assumed. Please refer to Appendix 1.A.2 for the proof. Figures ww are drawn by the simplied formula when si2 = 0.55, β22 = 0.05. 9 The simplied formula for ηi derived in Appendix 1.A.2 clearly shows that ηi is a function of ξj s given the actual cost shares and parameters. 9 mization error. Moreover, ηi is derived based on the assumption that ξj s do not vary across producers, so it would be unnatural to interpret ηi as a stochastic component of the cost share equations. If the assumption about ξj s is relaxed as described in the previous para- graph, the term representing allocative ineciency in the cost share equations can be seen 10 as a stochastic component. However, as Reiss and Wolak (2007) point out , there are other sources of random components in the cost share equations besides failure in cost minimiza- tion. Furthermore, as noted by Brown and Walker (1995), one can easily make distributional assumptions of the system and apply usual estimators by using additive noise terms for the share equations. The second and third points are related to the issue on the stochastic specication in the models of producers' demand, cost, and production systems. There are contradictory views on how to formulate a stochastic specication for the cost system. The conventional practice is to append additive noise terms to the nonstochastic cost share equations. For example, Christensen and Greene (1976) state that since the cost share equations are derived by dierentiation, they do not contain the disturbance term from the cost function (p.662), so they add stochastic noise terms following multivariate normal distribution to the cost share equations in an ad hoc fashion. Subsequent research criticizes that such an approach is inconsistent with economic theory and derives stochastic components in the cost share equations in the optimization framework. These studies include Chavas and Segerson (1987), McElroy (1987), Brown and Walker (1995), and Kumbhakar and Tsionas (2011). Note that although they provide theoretical justications for the stochastic specication of the cost share equations, the sources of stochastic components in the cost share equations vary across studies, such as random environments, measurement errors, and optimization errors. In order to deal with these issues, I modify the following assumptions to Kumbhakar (1997)'s model. 10 The four principal ways in which a researcher can introduce stochastic components into a deterministic economic model are: 1. researcher uncertainty about the economic environment; 2. agent uncertainty about the economic environment; 3. optimization errors on the part of economic agents; and 4. measurement errors in observed variables. (p.4305) 10 Assumption 1. The magnitudes of allocative ineciency vary across producers and are denoted by ξ , · · · , ξ . i2 iJ Assumption 2. uTi is uncorrelated with ξ i2 , · · · , ξiJ but positively correlated with |ξ |, · · · , i2 |ξiJ |. Assumption 3. Additive noise terms, denoted by ν i = (νi2 , · · · , νiJ ), are allowed in the cost share equations. As mentioned, Assumptions 1 and 2 are made to introduce the idiosyncratic disturbance due to optimization errors, as well as to precisely model dependence between technical and allocative ineciency. Kumbhakar (1997)'s model includes several terms induced by alloca- tive ineciency, including ξj s and ηi. Thus, instead of modifying the assumption on the magnitudes of allocative ineciency, we can assume that uTi is uncorrelated with ηi2 , · · · , ηiJ but positively correlated with |ηi2 |, · · · , |ηiJ |. In this case, ξj s are not used for estimation. This approach is similar to Schmidt and Lovell (1980) and Amsler et al. (2021), but there are mainly two reasons for imposing Assumption 1 other than ξj or ξij being the origin of allocative ineciency in the model. First of all, the alternative method does not provide a solution to the Greene Problem. The specication of the two previous studies, Schmidt and Lovell (1980) and Amsler et al. (2021), follows Schmidt and Lovell (1979) as yi = α + x′i β + vi − uTi   β1 wij xi1 − xij = ln + eij , j = 2, ..., J, βj wi1 where yi is the natural log of output of producer i, xi is a vector of natural log of inputs, vi ∈ R is a random distrubance, uTi ∈ R+ represents technical ineciency, wij ∈ R++ is the price of input j, and eij ∈ R is a two-sided term capturing allocative ineciency and noise. This model does not include an additional term representing allocative ineciency in the production frontier, therefore issues like the Greene Problem, which occur in the 11 translog cost system, are not raised. However, an analytic solution, which is proposed by Kumbhakar (1997) and Kumbhakar and Tsionas (2005a,b) that introduce ξj s and make a distributional assumption on them, is not applicable if ξj s are not used when estimating the translog cost system. In addition, even though both uA i and ηi are functions of ξj s, no closed-form expression for uA i in terms of ηi exists in the translog cost system. In other words, uAi cannot be specied as a function of ηi. Thus, a method similar to approximation ′ solutions proposed by Schmidt (1984) that specify a functional relationship as uAi = ei Aei , where A is a positive semi-denite matrix, cannot be applied as well. Secondly, it should be noted that ηi might not correctly measure each rm's degree of allocative ineciency in the model, although it arises from the fact that a producer uses inputs in an incorrect proportion. Figure 1.2.2(b) shows that the absolute value of the error term in the cost share equation, |ηi2 |, can decrease as the size of |ξ2 | increases. This indicates that a producer using inputs fairly ineciently can have the same cost shares to a producer that allocates inputs eciently. In addition, it is not guaranteed that ξj = 0 implies ηij = 0 by construction, which implies that the input j 's actual share deviates from its optimum share even if the input pair (j, 1) is eciently allocated. For instance, suppose that J = 3, ξ2 = 0, but ξ3 ̸= 0. Then, although the input pair (2,1) is eciently allocated, ηi2 ̸= 0 because of the presence of ineciency among the input pair (3,1). Consequently, the magnitudes of allocative ineciency are allowed to vary and linked to uTi , which represents the magnitudes of technical ineciency, in order to rene Kumbhakar (1997)'s model and to provide a solution to the Greene Problem. I further discuss As- sumption 2, which is about how to model the dependence between uTi and ξij s. Since the terms representing technical and allocative ineciency are introduced to the model without any theoretical linkages, uTi is assumed to be uncorrelated with ξi2 , · · · , ξiJ . However, as illustrated in Figure 1.2.1, a producer becomes technically inecient as the size of uTi , which is non-negative, increases and becomes allocatively inecient as the size of |ξij | increases. Hence, uTi is assumed to be positively correlated with |ξi2 |, · · · , |ξiJ |. 12 Assumption 3 is imposed for two purposes. The rst purpose is to capture not only opti- mization errors stemmed from ξij s in the model but also sources of randomness that are not explicitly modeled. For example, Kumbhakar and Tsionas (2005a) use the same specica- 11 tion to take account of measurement errors, and agent and/or researcher uncertainty. The second purpose is to facilitate estimation. As pointed out, it is dicult to derive the distri- bution of ηi from ξij s. Moreover, although the assumption on the magnitudes of allocative inecency is modied from Kumbhakar (1997)'s model, the cost share equations can be seen as deterministic because ηij is a function of ξij s. By appending additive noise, the system is converted to a stochastic model so that one can readily obtain a joint density for estimation. The rst and second purposes are related. Reiss and Wolak (2007), for instance, state that one can simply transform a deterministic economic model into an econometric model and justify applying usual estimators by introducing measurement errors. 1.2.2.2 Modied Model and Distributional Assumptions Because of Assumptions 1 and 3, the stochastic cost frontier model needs to be modied. Although the assumption about the magnitudes of allocative ineciency has changed from Kumbhakar (1997)'s model, the formula for each component of the system can be identically sj (y i , wi ) = 1 and Jj=1 ηij = 0 ∀i, an PJ P derived. However, since j=1 additional restriction PJ on the sum of νij is required so that j=1 sij = 1 ∀i is guaranteed. The modied model can be written as lnCi = lnC(y i , wi ) + ϵi = lnC(y i , wi ) + vi + ui = lnC(y i , wi ) + vi + uTi + uA i = lnC(y i , wi ) + vi + uTi + g(ξ i ) 11 These error terms represent measurement error and/or factors that are not under the control of the rm, so they are not modeled explicitly, unlike the ξ 's. Alternatively, these errors might not be relevant for the producer (in the sense that they are known to the producer), but nonetheless must be taken into account by the researcher (who does not know them). (p.739) 13 sij = sj (y i , wi ) + eij = sj (y i , wi ) + ηij + νij = sj (y i , wi ) + hj (ξ i ) + νij , j = 2, · · · , J. (1.2.1) Recall that Ci is the actual cost of producer i, C(y i , wi ) is the deterministic kernel of the stochastic cost frontier, y i ∈ RM + is a vector of M outputs, wi ∈ RJ++ is a vector of input prices, vi ∈ R is a random disturbance, uTi ∈ R+ represents a cost increase due to technical ineciency, uAi = g(ξ i ) ∈ R+ represents a cost increase due to allocative ineciency, ξ i = (ξi2 , · · · , ξiJ ), ξij , j = 2, · · · , J, represents allocative ineciency for the input pair (j, 1), ui = uTi + uA i , ϵi = vi + ui , sij ∈ [0, 1] is the actual cost share of input j , sj (y i , wi ) ∈ [0, 1] is the optimum cost share of input j , ηij = hj (ξ i ) ∈ R is the disturbance due to allocative ineciency, νij ∈ R is additive noise, and eij = ηij + νij is the disturbance due to both allocative ineciency and noise. Each component of the system can be written as M M M X y 1 X X yy lnC(y i , w i ) = β0 + βm (lnyim ) + β (lnyim )(lnyin ) m=1 2 m=1 n=1 mn J J J X 1 X X ww + βjw (lnwij ) + β (lnwij )(lnwik ) j=1 2 j=1 k=1 jk XM X J yw + βmj (lnyim )(lnwij ) m=1 j=1 XJ XM yw sj (y i , wi ) = βjw + ww βjk (lnwik ) + βmj (lnyim ), j = 2, · · · , J k=1 m=1 J J X J J J X X 1 X X ww uAi = g(ξ i ) = βjw ξij + ww βjk (lnwij )ξik + β ξij ξik j=1 j=1 k=1 2 j=1 k=1 jk XM XJ X J yw + βmj (lnyim )ξij + ln (s∗ij /eξij ) m=1 j=1 j=1 PJ PJ sj (y i , wi )[1 − { k=1 (s∗ik /eξik )}eξij ] + k=1 ww βjk ξik ηij = hj (ξ i ) = , j = 2, · · · , J, (1.2.2) { Jk=1 (s∗ik /eξik )}eξij P PJ where s∗ij = sj (y i , wi ) + k=1 ww βjk ξik , j = 2, · · · , J, is the shadow cost share of input j for PJ producer i who is assumed to be only allocatively inecient. In addition, j=1 νij = 0 ∀i 14 PJ is required to guarantee j=1 sij = 1 ∀i. Restrictions on parameters are also necessary yy yy in order to satisfy the properties of the cost function such that βmn = βnm ∀m ̸= n and = 0 ∀j , and Jj=1 βmj PJ PJ ww yw ww ww βjw = 1, P βjk = βkj ∀j ̸= k for symmetry, j=1 k=1 βjk = 0 ∀m for linear homogeneity in wi . Regarding the distributional assumptions on the stochastic components of the system, I follow a standard practice, such as in Christensen and Greene (1976), Schmidt and Lovell (1979), and Kumbhakar and Tsionas (2005a,b). Assume that vi is distributed as N (0, σv2 ), uTi is distributed as |N (0, σT2 )|, ξ i is distributed as N (0, Σξ ), and νi is distributed as N (0, Σν ). For j = 2, · · · , J , assume that uTi and ξij are distributed independently of vi and νij , and vi and νij are mutually independent. 1.3 Estimation Strategy 1.3.1 Relationship between uTi and ξi : APS Copulas In Section 1.2.2.1, Assumption 2, which is about the dependence between uTi and ξi , is imposed in order to have the desirable attributes between technical and allocative ineciency. Based on this assumption, it is required to derive the joint density of uTi and ξ i to estimate the model. One way to obtain the joint distribution of dependent random variables is applying copulas; that is, given specic marginal distributions of uTi and ξij s, the joint distribution of them can be obtained by employing copulas that capture the dependence. Sklar's theorem states that for every joint cumulative distribution function of random variables X1 , · · · , XJ with margins F1 (·), · · · , FJ (·), which are marginal cumulative distribu- tion functions of X1 , · · · , XJ , there exists a copula C : [0, 1]J → [0, 1], which is a cumulative distribution function, such that  F (x1 , · · · , xJ ) = C F1 (x1 ), · · · , FJ (xJ ) for all xi ∈ R, i = 1, · · · , J , where F (·) is a joint cumulative distribution function. 15 Let ω1 = F1 (uT ), ω2 = F2 (ξ2 ), · · · , ωJ = FJ (ξJ ), where uT , ξ2 , · · · , ξJ are dummy argu- ments. In order to have the desired properties between technical and allocative ineciency, it is required that ω1 is linked to ω2 , · · · , ωJ , for which ω1 is uncorrelated with ω2 , · · · , ωJ but correlated with |ω2 − 0.5|, · · · , |ωJ − 0.5|. To be specic, ω1 is uncorrelated with ω2 , · · · , ωJ so that uTi is uncorrelated with ξi2 , · · · , ξiJ . However, as Figure 1.3.1 shows, if we assume that ξij is distributed symmetric around zero, like ξij ∼ N (0, σξj ), a producer becomes al- locatively inecient when ωj , j = 2, · · · , J , moves away from 0.5. Therefore, ω1 needs to be positively correlated with |ω2 − 0.5|, · · · , |ωJ − 0.5|. Figure 1.3.1: PDF and CDF of ξij (a) PDF of ξij (b) CDF of ξij Amsler et al. (2021) propose a new family of copulas (hereafter the APS copulas) that can induce the desired attributes between technical and allocative ineciency. For example, suppose that two inputs are used for production (J = 2). Then, the APS-2 copulas can be applied to capture dependence between uTi and ξi2 . The APS-2 copula densities are dened as follows (Amsler et al., 2021, p.4): APS-2-A : c12 (ω1 , ω2 ) = 1 + θ12 (1 − 2ω1 ){1 − 12(ω2 − 0.5)2 )}, |θ12 | ≤ 0.5 APS-2-B : c12 (ω1 , ω2 ) = 1 + θ12 (1 − 2ω1 )(1 − 4|ω2 − 0.5|), |θ12 | ≤ 1, where c12 (ω1 , ω2 ) is the APS-2 copula density, and θ12 is the association parameter. Then, cov(ω1 , ω2 ) = 0, corr(ω1 , (ω2 −0.5)2 ) = √2 θ for the APS-2-A copula, and corr(ω1 , |ω2 −0.5|) = 15 1 3 θ for the APS-2-B copula (Amsler et al., 2021, p.3-4); that is ω1 is uncorrelated with ω2 , but 16 it can be correlated with |ω2 − 0.5|. Given this, uTi and ξi2 can have the desired properties. Figure 1.3.2 illustrates the sample correlation between ω1 , ω2 , and (ω2 − 0.5)2 when θ12 = 0.4 derived from simulations 12 using the APS-2-A copula. It shows that corr(ω1 , ω2 ) ≈0 and corr(ω1 , (ω2 − 0.5)2 ) ≈ 0.2066 as the theoretical results. Figure 1.3.2: Sample Correlations (The APS-2-A Copula) (a) corr(ω1 , ω2 ) (b) corr(ω1 , (ω2 − 0.5)2 ) Amsler et al. (2021) also develop a method that the APS-2 copulas can be extended to more dimensions, which is necessary when more than two inputs are used for production. If J = 3, for instance, the APS-3 copulas can be applied to capture dependence between uTi , ξi2 , and ξi3 . To be specic, ω2 and ω3 are allowed to follow any standard bivariate copula but need to be linked to ω1 as in the APS-2 copulas. Assume that ω2 and ω3 follow the bivariate Gaussian copula. Then, the APS-3 copula densities are dened as follows (Amsler et al., 2021, p.6): APS-3-A : c123 (ω1 , ω2 , ω3 ) = 1 + (c12 − 1) + (c13 − 1) + (c23 − 1), where c12 = c12 (ω1 , ω2 ) = 1 + θ12 (1 − 2ω1 ){1 − 12(ω2 − 0.5)2 )}, |θ12 | ≤ 0.5 c13 = c13 (ω1 , ω3 ) = 1 + θ13 (1 − 2ω1 ){1 − 12(ω3 − 0.5)2 )}, |θ13 | ≤ 0.5 1 h ρ2 Φ−1 (ω )2 − 2ρΦ−1 (ω )Φ−1 (ω ) + ρ2 Φ−1 (ω )2 i 2 2 3 3 c23 = c23 (ω2 , ω3 ) = p exp − . 2 2(1 − ρ 2) 1−ρ 12 The number of replications is 1,000, where the sample size is 1,000 for each replication. 17 APS-3-B : c123 (ω1 , ω2 , ω3 ) = 1 + (c12 − 1) + (c13 − 1) + (c23 − 1), where c12 = c12 (ω1 , ω2 ) = 1 + θ12 (1 − 2ω1 )(1 − 4|ω2 − 0.5|), |θ12 | ≤ 1 c13 = c13 (ω1 , ω3 ) = 1 + θ13 (1 − 2ω1 )(1 − 4|ω3 − 0.5|), |θ13 | ≤ 1 1 h ρ2 Φ−1 (ω )2 − 2ρΦ−1 (ω )Φ−1 (ω ) + ρ2 Φ−1 (ω )2 i 2 2 3 3 c23 = c23 (ω2 , ω3 ) = p exp − , 2 2(1 − ρ ) 2 1−ρ where c123 (ω1 , ω2 , ω3 ) is the APS-3 copula density, c12 (ω1 , ω2 ) and c13 (ω1 , ω3 ) are the APS-2 copula densities, c23 (ω2 , ω3 ) is the bivariate Gaussian copula density, θ12 and θ13 are the association parameters, Φ(·) is the cumulative distribution function of the standard normal distribution, and ρ ∈ (−1, 1) is the correlation parameter. By applying the APS-3 copulas, one can capture dependence such that ω1 is uncorrelated with ω2 and ω3 but correlated with |ω2 − 0.5| and |ω3 − 0.5| as in the case of J = 2. Hence, uTi and ξij s, j = 2 and 3, can have the desired properties. Figure 1.B.1 illustrates the sample correlations between ω1 , ω2 , (ω2 − 0.5)2 , ω3 , and (ω3 − 0.5)2 when θ12 = θ13 = 0.2 and ρ = −0.5 13 obtained by 14 simulations based on the APS-3-A copula. It shows that corr(ω1 , ω2 ) and corr(ω1 , ω3 ) are approximately zero when J = 2, but corr(ω1 , |ω2 − 0.5|) and corr(ω1 , |ω3 − 0.5|) are positive. 1.3.2 Derivation of the Joint Density By employing the APS copulas, we can derive the joint density of the translog cost system that captures the dependence between technical and allocative ineciency. For notational convenience, rewrite X = ϵi = vi + uTi + uA i = X1 + Z1 + g(Z 2 ), Y = (Y2 , · · · , YJ ) = (ei2 , · · · , eiJ ) = (ηi2 + νi2 , · · · , ηiJ + νiJ ) = (h2 (Z 2 ) + W2 , · · · , hJ (Z 2 ) + WJ ), and Z = (Z1 , Z 2 ) = (uTi , ξi2 , · · · , ξiJ ). Let θ = (θ 1 , θ 2 , θ 3 ) be a vector of parameters, where θ1 = 13 Amsler et al. (2021) show that the allowable range of θ12 and θ13 depends on ρ. To be specic, if ω2 and ω3 are strongly correlated, the range of |θ12 | + |θ13 | decreases. Please refer to Result 10 of Amsler et al. (2021, p.6) in detail. 14 The number of replications is 1,000, where the sample size is 1,000 for each replication. 18 (β0 , β y , β yy , β w , β ww , β yw , σv2 , Σν ), θ 2 = (θ APS ,θ Gauss ), and θ 3 = (σT2 , Σξ ). 15 θ APS and θ Gauss denote vectors of association or correlation parameters of the APS and Gaussian copulas, 16 respectively. Then, the translog cost system can be rewritten as lnCi = lnC(y i , wi ) + X = lnC(y i , wi ) + X1 + Z1 + g(Z 2 ) sij = sj (y i , wi ) + Yj = sj (y i , wi ) + hj (Z 2 ) + Wj , j = 2, · · · , J. (1.3.1) It is required to derive the joint density of X = ϵi and Y = (Y2 , · · · , YJ ) = (ei2 , · · · , eiJ ) in order to form a likelihood function of the translog cost system. First, the joint density of X, Y , and Z can be written as fX,Y ,Z (x, y, z; θ) = fX,Y |Z (x, y|z; θ) · fZ (z; θ 2 , θ 3 ) = fX|Y ,Z (x|y, z; θ) · fY |Z (y|z; θ) · fZ (z; θ 2 , θ 3 ). Then, the joint density of X and Y can be obtained by integrating out Z such that Z fX,Y (x, y; θ) = fX,Y ,Z (x, y, z; θ)dz R+ ×RJ−1 Z = fX|Y ,Z (x|y, z; θ) · fY |Z (y|z; θ) · fZ (z; θ 2 , θ 3 )dz. (1.3.2) R+ ×RJ−1 Suppose that the distribution of Z is known and simple. Then, the joint density of X and Y (1.3.2) can be approximated by drawing random numbers from the density of Z. APS Gauss However, θ 2 = (θ ,θ ), which governs the joint distribution of Z , should be estimated. This implies that we need to transform the random vector Z to another random vector in order to estimate θ2 . In addition, a transformed random vector needs to be simple to make 17 the process of drawing random numbers easy. 15 Notation in bold represents row vectors whose elements are corresponding parameters of the system. For instance, β y = (β1y , β2y , · · · ) and β yy = (β11 yy yy , β12 yy , · · · , β21 yy , β22 , · · · ). 16 For APS Gauss example, if J = 3, θ = (θ12 , θ13 ) and θ = ρ. 17 If the researcher wants to take a draw from a standard normal density (that is, a normal with zero mean and unit variance) or a standard uniform density (uniform between 0 and 1), the process from a programming perspective is very easy. (Train, 2009, p.205-206) 19 As pointed out in Section 1.2.2.1, it is dicult to apply the change of variables theorem into the Kumbhakar (1997)'s model when the magnitudes of allocative ineciency are as- sumed to be random. However, the theorem can be used to derive the joint density of the translog cost system under the assumptions made in Section 1.2.2.1 by the following proce- dure. It is established on the probability integral transform and the copula-based version of Rosenblatt transformation (see Rosenblatt, 1952) that are monotone. Rosenblatt (1952) proposes a method using conditional cumulative distribution func- tions for transforming a dependent random vector to the independent random vector whose components are uniformly distributed on [0, 1]. Appendix 1.A.3 provides detailed explana- tions for the Rosenblatt transformation. Based on the both transformations, Z that has dependent components can be replaced with functions of the independent ramdom vector ζ = (ζ1 , · · · , ζJ ), where ζj ∼ U [0, 1], j = 1, · · · , J are uniformly and independently dis- tributed over [0, 1]. The procedure for a change of variables using the probability integral transform and the Rosenblatt transformation is illustrated in Figure 1.3.3. Figure 1.3.3: Procedure for a Change of Variables The rst step is to replace Z with functions of ω using the probability integral transform. Given that ωj = F (zj ; θ3 ) for j = 1, · · · , J , let JZ be the Jacobian matrix whose (i, j)th 20 ∂zi ∂Fi−1 (ωi ;θ3 ) element is ∂ωj = ∂ωj . Then, by applying the change of variables theorem, the joint density of X and Y can be written as fX,Y (x, y; θ) Z    = fX|Y ,Z x|y, z(ω); θ · fY |Z (y|z ω); θ · fZ z(ω); θ 2 , θ 3 · |JZ |dω [0,1]J Z Y J    = fX|Y ,Z x|y, z(ω); θ · fY |Z y|z(ω); θ · c(ω1 , · · · , ωJ ) · fj zj (ωj ) · |JZ |dω [0,1]J j=1 Z   = fX|Y ,Z x|y, z(ω); θ · fY |Z y|z(ω); θ · c(ω1 , · · · , ωJ )dω, [0,1]J where ω = (ω1 , · · · , ωJ ). The second equality holds as a joint density of dependent ran- dom variables equals the product of the copula density c(ω1 , · · · , ωJ ) and marginal densities  fj zj (ωj ) , j = 2, · · · , J, for each random variable. The third equality holds as     ∂z1 ∂z1 ∂z1 1  ∂ω1 ∂ω2 ··· ∂ωJ   f1 (z1 ;θ3 ) 0 ··· 0   ∂z ∂z2 ∂z2     1  2  ∂ω ∂ω2 · · · ∂ω  0 f2 (z2 ;θ 3 ) ··· 0  JZ =  . 1 J = ,   . .. .  . . ..    .. . . . . .   . . . . . 0         ∂zJ ∂zJ ∂zJ 1 ∂ω1 ∂ω2 · · · ∂ω J 0 0 ··· fJ (zJ ;θ 3 ) hQ J i−1 thus |JZ | = j=1 fj zj ; θ 3 . The second step is to replace ω with functions of ζ employing the Rosenblatt trans- formation. Considering that a copula C is also a cumulative distribution function, let TR : [0, 1]J → [0, 1]J be the Rosenblatt transformation given by ζ1 = C1|2,··· ,J (ω1 |ω2 , · · · , ωJ ) . . . ζJ−2 = CJ−2|J−1,J (ωJ−2 |ωJ−1 , ωJ ) ζJ−1 = CJ−1|J (ωJ−1 |ωJ ) ζJ = CJ (ωJ ). For example, the conditional APS-2-A copula function C1|2 (ω1 |ω2 ) is C1|2 (ω1 |ω2 ) = ω1 + g2 ω1 (1 − ω1 ), 21 where g2 = θ12 {1−12(ω2 −0.5)2 }. The conditional APS-3-A copula functions C1|23 (ω1 |ω2 , ω3 ) and C2|3 (ω2 |ω3 ) are 1 C1|23 (ω1 |ω2 , ω3 ) = {h(ω1 − ω12 ) + c23 ω1 } c23  Φ−1 (ω ) − ρΦ−1 (ω )  2 3 C2|3 (ω2 |ω3 ) = Φ p , 1−ρ 2 where c23 = c23 (ω2 , ω3 ) is the bivariate Gaussian copula density, h = g2 + g3 , where g2 = θ12 {1 − 12(ω2 − 0.5)2 } and g3 = θ13 {1 − 12(ω3 − 0.5)2 }, Φ is the cumulative distribution function of the standard normal distribution, and ϕ is the probability density function of the 18 standard normal distribution. th Let T be the inverse function of TR , and JT denotes the Jacobian matrix whose (i, j) ∂ωi element is . Then, by applying the change of variables theorem once more, the joint ∂ζj density of X and Y can be written as fX,Y (x, y; θ) Z        = fX|Y ,Z x y, z T (ζ) ; θ · fY |Z y z T (ζ) ; θ · c T1 (ζ), · · · , TJ (ζ); θ2 |JT |dζ [0,1]J Z       = fX|Y ,Z x y, z T (ζ) ; θ · fY |Z y z T (ζ) ; θ dζ [0,1]J where Tj (ζ) = ωj , j = 1, · · · , J . The second equality holds as     ∂ω1 ∂ω1 ∂ω1 1 1 1 ··· ···  ∂ζ1 ∂ζ2 ∂ζJ   c1|2,··· ,J ∂C1|2,··· ,J /∂ωJ−1 ∂C1|2,··· ,J /∂ωJ  . . .. .   . .. . . . . .   .. . .    . . . . . . .  JT =  = ,      ∂ωJ−1 ∂ωJ−1 · · · ∂ω J−1   0 ··· 1 1   ∂ζ1 ∂ζ2 ∂ζJ   cJ−1|J ∂CJ−1|J /∂ωJ      ∂ωJ ∂ωJ ∂ζ1 ∂ζ2 · · · ∂ω ∂ζJ J 0 ··· 0 1 cJ where c1|2,··· ,J , · · · , cJ−1|J are conditional copula densities. Therefore, |JT | = [c1|2,··· ,J × · · · × cJ−1|J × cJ ]−1 = [c(ω1 , · · · , ωJ )]−1 . 19 18 Sections 2.A.1 and 2.2 provide derivation of these conditional copula functions. 19 Arguments are occasionally dropped for brevity. 22 Note that    fX|Y ,Z x y, z T (ζ) ; θ           = fX|Y ,Z x1 + z1 T (ζ) + g z 2 T (ζ) h2 z 2 T (ζ) , · · · , hJ z 2 T (ζ) , z T (ζ) ; θ      = fX1 x − z1 T (ζ) − g z 2 T (ζ) ; θ , where x = lnC − lnC(y, w), and    fY |Z y z T (ζ) ; θ        = fY |Z h2 z 2 T (ζ) + ν2 , · · · , hJ z 2 T (ζ) + νJ z T (ζ) ; θ       = fν e2 − h2 z 2 T (ζ) , · · · , eJ − hJ z 2 T (ζ) ; θ , where ej = sj − sj (y, w), j = 2, · · · , J . Therefore, the joint density of X and Y can be simplied as fX,Y (x, y; θ) Z       = fX|Y ,Z x y, z T (ζ) ; θ · fY |Z y z T (ζ) ; θ dζ [0,1]J Z      = fX1 x − z1 T (ζ) − g z 2 T (ζ) ; θ [0,1]J      · fν e2 − h2 z 2 T (ζ) , · · · , eJ − hJ z 2 T (ζ) fζ (ζ)dζ       = Eζ fX1 x − z1 T (ζ) − g z 2 T (ζ) ; θ       · fν e2 − h2 z 2 T (ζ) , · · · , eJ − hJ z 2 T (ζ) ; θ , (1.3.3) where fζ (ζ) is the joint probability density function of ζ, and Eζ represents the expectation with respect to the distribution of ζ. The second equality holds as ζ is the independent random vector such that each component follows a uniform distribution over [0, 1], which implies fζ (ζ) = 1. 23 1.3.3 Maximum Simulated Likelihood Estimator Since the joint density of X and Y involves an intractable integral, simulation-based methods are necessary to compute the joint density. The direct or crude frequency simulator for the joint density fX,Y (x, y; θ) can be written as fˆX,Y (x, y; θ) R    1X (r)   (r)  = fX1 x − z1 T (ζ ) − g z 2 T (ζ ) ; θ R r=1    (r)   (r)  · fν e2 − h2 z 2 T (ζ ) , · · · , eJ − hJ z 2 T (ζ ) ; θ , (r) (r) where ζ (r) = (ζ1 , · · · , ζ1 ) is the independent r th draw of R draws from multivariate stan- MSL dard uniform distribution. The maximum simulated likelihood estimator θ̂ maximizes the following simulated log likelihood function: lnL̂(θ) X N = lnfˆX,Y (x, y; θ) i=1 N  X R    X 1 (r)   (r)  = ln fX1 x − z1 T (ζ ) − g z 2 T (ζ ) ; θ i=1 R r=1     (r)  (r)  · fν e2 − h2 z 2 T (ζ ) , · · · , eJ − hJ z 2 T (ζ ) ; θ . (1.3.4) 1.4 Firm-level Ineciency Researchers can measure and decompose the average ineciency by estimating the model proposed in previous sections. However, as Jondrow et al. (1982) point out, it is also desirable to estimate ineciency for each observation to compare (in)eciency levels across rms, which is the original motivation of Farrell (1957). However, the error term in the cost share equations, eij , is assumed to contain both the disturbance from the optimal share and the additive noise. Therefore, methods that use conditional distributions to derive rm-level ineciency developed in previous studies, such as Jondrow et al. (1982) and Battese and 24 Coelli (1988), cannot be directly employed. In this section, I propose a strategy to measure and decompose rm-level technical and allocative ineciency for (1.2.1). The process unfolds in four steps as summarized in Figure 1.4.1. The rst step is de- composing the error term in the cost share equations, eij , into ηij and νij to calculate η̂ij . As a point estimate of η̂ij , we can use the mode of the conditional distribution fˆη|e (η|e) as Jondrow et al. (1982), which can be obtained by deconvolution density estimation. The second step is solving for ξi given η̂ij . As J − 1 cost share equations ηij = hj (ξ i ), j = 2, · · · J , include J −1 unknowns, we can nd the solution to the system of equations, ξ̂ i . The third step is calculating uAi using ξ̂ i following (1.2.2). The last step is estimating the conditional expectation of uTi given ûA i and ϵ̂i , where ûTi = E[uTi |vi + uTi ] = E[uTi |ϵi − uA i ] that is in line with Jondrow et al. (1982). Figure 1.4.1: Process to Measure and Decompose Individual Ineciency Step 1: Decompose ηij and νij Step 2: Solve for ξi given η̂ij Step 3: Calculate uA i using ξ̂ i Step 4: Estimate uTi given ûA i Details on Step 1 and 4 are as follows. First, since the two terms representing allocative ineciency, ηij and uA i , are functions of ξ i , it is required to estimate ξ i to measure individual allocative ineciency, which can be obtained from the system of equations ηij = hj (ξ i ), j = 2, · · · , J . However, because the error term in the cost share equations, eij , which can be obtained after estimating the model, contains additive noise, νij , it is essential to decompose ηij and νij that are unobservable. Density deconvolution methods can be used to recover 25 an unknown probability density function that is noise-free, which implies we can recover the density function of ηij . The setup of the density deconvolution problem is as follows. Suppose that one can only observe samples Y1 , · · · , Yn given by Yi = Xi +Ui , i = 1, · · · , n, where Ui is noise from a known distribution and independent of Xi . The problem is how to estimate the density function of X , fX (x), and the conditional density of X given Y , fX|Y (x|y), based on the observations Y1 , · · · , Yn . For more details about density deconvolution to estimate fX (x), please refer to Carroll and Hall (1988), Stefanski and Carroll (1990), and Fan (1991). Wang and Ye (2015) propose re-weighted deconvolution kernel methods to estimate the conditional density function fX|Y (x|y) in an additive error model. Their estimator, which is applied to estimate fη|e (η|e), is Xn fˆX|Y (x|y) = τ̂0 (x|y) Kh∗ (x − Yj ), j=1 where τ̂0 (x|y) = PnfU (y−x) , Lb (·) = L(·/b)/b, L(·) is a real non-negative kernel function, b j=1 Lb (y−Yj ) is the bandwidth that associates with the kernel density estimate of fY , Kh∗ (·) = K ∗ (·/h)/h, K ∗ (z) = 1 e−itz ϕϕUK(t/h) (t) R 2π dt is the deconvoluting kernel, h ∈ R++ is a smoothing parameter, R ϕU is the characteristic function of U, and ϕK (t) = eitχ K(χ)dχ is the Fourier transform of K(χ). Then, ηij can be estimated from the estimates of fη|e (η|e). Second, upon solving for ξi given η̂ij (Step 2) and calculating uAi given ξˆi by (1.2.2) (Step 3), one can obtain vi + uTi from the cost function. Then, an approach like Jondrow et al. (1982) and Battese and Coelli (1988), which use the conditional expectation, can be applied to estimate technical ineciency. As it is assumed that vi ∼ N (0, σv2 ), uTi ∼ |N (0, σT2 )|, conditional expectation of uTi given vi + uTi is ϕ( σλ ϵ̃i ) λ   E[uTi |vi + uTi ] = σ∗ + ϵ̃i Φ( σλ ϵ̃i ) σ where σ∗2 = σv2 σT2 /σ 2 , σ 2 = σv2 + σT2 , λ = σT + σv , ϕ is the probability density function of the standard normal distribution, Φ is the cumulative density function of the standard normal distribution, and ϵ̃i = vi + uTi (Kumbhakar and Lovell, 2000, p.141). 26 1.5 Application: U.S. Banking Industry 1.5.1 Data The new model and strategies are applied to the U.S. banking industry. The dataset for this empirical study is based on the Reports of Condition and Income (Call Reports) for all FDIC insured U.S. banks. These are retrieved from FDIC's Statistics on Depository Institutions (https://www7.fdic.gov/sdi). Although the sample includes all U.S. depository institutions in the Call Report for the end of 2019 and 2020, it is ltered for the following reasons. First, the observations are dropped if key variables in the model are missing. Second, banks that enter or exit the market during the corresponding year are excluded, as their reported cost does not represent the total yearly cost. To the end, the number of banks in the dataset is 5,105 in 2019 and 4,937 in 2020, whereas the number of reported institutions is 5,177 in 2019 and 5,002 in 2020, respectively. 20 There is a long-standing disagreement over the input and output of banks. I follow the asset (or intermediation) approach (see Sealey and Lindley, 1977) that banks are regarded as rms that transform various nancial and physical resources, such as deposits and labor, into loans and investments. Similar to Altunbas et al. (2007) and Ding and Sickles (2019), it is assumed that banks produce two outputs using three inputs. The output variables are loans (yi1 ) and other earning assets (yi2 ), such as securities and trading assets. The input variables are funds that the bank owes (xi1 ), such as deposits and debentures, the number of full-time employees (xi2 ), and xed assets (xi3 ). Given this denition of inputs, input prices are dened as wi1 = interest expenses (Ci1 )/funds that the bank owes (xi1 ), wi2 = salaries (Ci2 )/the number of full-time employees (xi2 ), and wi3 = xed assets expenses (Ci3 )/xed assets (xi3 ). The total cost (Ci ) is dened as the sum of interest expenses (Ci1 ), salaries (Ci2 ), and xed assets expenses (Ci3 ). Tables 1.B.1 and 1.B.2 summarize descriptive statistics of key variables in 2019 and 2020, 20 For more details, please refer to Berger and Humphrey (1992) and Guarda et al. (2013). 27 respectively. While other variables are not changed signicantly during the sample period, there are notable changes in two key variables as shown in Table 1.5.1. First, interest expenses decrease signicantly in 2020. Although they had had an increasing trend since 2016, they decreased from $158.7 billion in 2019 to $77.1 billion in 2020 for all depository institutions. Second, other earning assets have increased substantially during the pandemic. To be specic, their balance has slightly increased from 2016 to 2019 but increased from $6.9 trillion in 2019 to $9.8 trillion in 2020. It implies that the composition of banks' costs and outputs has considerably changed since the outbreak of COVID-19. This might lead the changes in the cost frontier of the U.S. banking industry between 2019 and 2020. Table 1.5.1: Interest Expenses and Other Earning Assets of the U.S. Banks Year 2016 2017 2018 2019 2020 Interest Expenses ($ billion) 54.4 73.3 119.8 158.7 77.1 Other Earning Assets ($ billion) 6,321.8 6,547.6 6,611.1 6,870.8 9,829.8 Source: FDIC 1.5.2 Average Ineciency The main purpose of this subsection is to check whether the average cost ineciency of the U.S. banking industry has varied before and after the pandemic. In addition, the estimation results are compared to those following Greene (1980) to show that more plausible results can be obtained from the model proposed in this paper. Greene (1980)'s assumptions can be summarized as follows. Given the translog cost system lnCi = lnC(y i , wi ) + ϵi = lnC(y i , wi ) + vi + ui sij = sj (y i , wi ) + eij , j = 2, 3, i.i.d i.i.d it is assumed that ϵi is independent of ei = (ei2 , ei3 ), vi ∼ N (0, σv2 ), ui ∼ |N (0, σu2 )|, and i.i.d ei ∼ N (0, Σe ). Maximum likelihood estimation can be applied to estimate parameters of 28 the model, where the joint density of ϵi , ei2 , ei3 is simply fϵ,e2 ,e3 (ϵ, e2 , e3 ) = fϵ (ϵ) · fe (e), as ϵi and ei are independent. Note that the probability density function of ϵi , which is the sum of two random variables following the normal distribution and the half normal distribution, respectively, is 2  ϵ   ϵλ  fϵ (ϵ) = ·ϕ ·Φ , σ σ σ where σ 2 = σv2 + σu2 , λ = σu /σv , and ϕ and Φ are density and distribution functions of the 21 standard normal distribution. Even though technical and allocative ineciency cannot be decomposed, overall ineciency ui can be estimated using Greene (1980)'s assumptions. Tables 1.5.2 and 1.5.3 show the estimation results for 2019 and 2020, respectively, where Models I and II stand for the model of this paper and the model following the independence assumption stated by Greene (1980). To estimate the model of this paper (Model I), R= 10, 000 sets of random numbers are drawn. Table 1.5.2: Estimation Results for 2019 Model I Model II Model I Model II Model I Model II β0 0.5176 0.5529 β11ww 0.1465 0.1488 σν2 /σe2 0.0625 0.0755 (0.1466) (0.1484) (0.0023) (0.0011) (0.0008) (0.0006) β1y 0.5130 0.5099 β12ww -0.1370 -0.1358 σν3 /σe3 0.0218 0.0431 (0.0178) (0.0187) (0.0020) (0.0010) (0.0010) (0.0004) β2y 0.4879 0.4883 β22ww 0.1365 0.1325 ρν /ρe 0.5233 0.0314 (0.0231) (0.0231) (0.0020) (0.0014) (0.0279) (0.0133) yy β11 0.1180 0.1147 β11yw 0.0088 0.0044 θ12 0.3229 - (0.0026) (0.0027) (0.0016) (0.0018) (0.1566) yy β12 -0.1102 -0.1069 β12yw -0.0060 -0.0025 θ13 0.0000 - (0.0031) (0.0030) (0.0012) (0.0014) (0.1506) yy β22 0.1005 0.0972 β21yw 0.0156 0.0187 ρθ 0.8137 - (0.0039) (0.0039) (0.0014) (0.0016) (0.0165) β1w 0.3704 0.3896 β22yw -0.0172 -0.0199 σT /σu 0.2242 0.2239 (0.0138) (0.0182) (0.0012) (0.0013) (0.0175) (0.0128) β2w 0.5368 0.5254 σv 0.2868 0.2922 σξ2 0.5941 - (0.0111) (0.0142) (0.0061) (0.0049) (0.0240) σξ3 0.6447 - (0.0228) Note: BHHH standard errors are in parentheses. 21 Please refer to Appendix 1.A.4 for the derivation. 29 Table 1.5.3: Estimation Results for 2020 Model I Model II Model I Model II Model I Model II β0 0.8529 0.9687 ww β11 0.1265 0.1276 σν2 /σe2 0.0553 0.0724 (0.1514) (0.1539) (0.0021) (0.0010) (0.0008) (0.0006) β1y 0.3599 0.3549 ww β12 -0.1224 -0.1202 σν3 /σe3 0.0201 0.0467 (0.0195) (0.0179) (0.0018) (0.0010) (0.0009) (0.0004) β2y 0.5805 0.5748 ww β22 0.1272 0.1222 ρν /ρe 0.3968 -0.1413 (0.0242) (0.0238) (0.0018) (0.0013) (0.0411) (0.0122) yy β11 0.1050 0.1053 yw β11 0.0086 0.0033 θ12 0.3191 - (0.0030) (0.0030) (0.0012) (0.0016) (0.2036) yy β12 -0.0841 -0.0846 yw β12 -0.0039 0.0012 θ13 0.0000 - (0.0034) (0.0030) (0.0011) (0.0013) (0.1428) yy β22 0.0647 0.0661 yw β21 0.0118 0.0159 ρθ 0.8162 - (0.0040) (0.0036) (0.0012) (0.0015) (0.0132) β1w 0.3633 0.3800 yw β22 -0.0180 -0.0219 σT /σu 0.2060 0.1690 (0.0103) (0.0147) (0.0011) (0.0013) (0.0269) (0.0227) β2w 0.5569 0.5429 σv 0.3193 0.3325 σξ2 0.6299 - (0.0088) (0.0117) (0.0069) (0.0058) (0.0227) σξ3 0.7005 - (0.0180) Note: BHHH standard errors are in parentheses. There are three points to be noted. First, the results indicate a dependence between technical and allocative ineciency, although θ13 , which captures dependence between uTi and ξi3 , is close to zero in both periods. Second, reecting large increase in other earning assets (yi2 ), β1y decreased and β2y increased in 2020 relative to 2019. Lastly, both σT of Model I and σu of model II decreased, implying that technical or overall ineciency decreased during the pandemic. This might be due to the fact that interest expenses decreased in 2020. However, the magnitude of decreases is much higher for Model II than for Model I. Table 1.5.4 shows the estimation results of average ineciency. In both models, the mean of uTi , uAi , or ui cannot be directly estimated; only their standard deviations and the standard deviation of ξij s can be estimated. However, since the mean of a random variable 22 that follows the half normal distribution is a function of its standard deviation , the mean of uTi for Model I and ui for Model II, which are assumed to follow the half normal distribution, can be calculated. The average uAi of Model I is calculated using the estimation results for √ 22 If X ∼ |N (0, σX 2 )|, E[X] = σX √ 2. π 30 rm-level ineciency in the next subsection. For Model I, the average ui is computed as the sum of the mean of both uTi and uA i . Table 1.5.4: Average Ineciency 2019 2020 ui uTi uAi ui uTi uA i Model I 0.2040 0.1789 0.0251 0.1887 0.1643 0.0244 Model II 0.1786 - - 0.1349 - - The results suggest that costs of U.S. banks increased by around 16∼18% and 2.5% during the sample period due to technical and allocative ineciency, respectively. There are two main ndings from the results. The rst one is that overall ineciency in 2020 decreased compared to 2019. The eect can be also decomposed. Cost increases due to allocative ineciency do not dier between 2019 and 2020 (2.5% → 2.4%), whereas changes in technical ineciency are non-trivial (17.9% → 16.4%). The second main nding is that assuming independence between ϵi and ei would produce unrealistic results. Particularly, the overall ineciency levels are generally underestimated by 3 ∼ 5% when compared to Model I. Furthermore, overall ineciency decreased to a great extend in 2020 in spite of the pandemic (17.9% → 13.5%), which is seemingly less plausible. 1.5.3 Firm-level Ineciency As the rst step to estimate rm-level ineciency, it is suggested in Section 1.4 that the mode of the conditional distribution fˆη|e (η|e) is used. Figures 1.5.1 and 1.5.2 show density estimates of e, fˆe (e), and conditional density estimates of η given e, fˆη|e (η|e), for 2019 obtained by kernel density estimation and density deconvolution. Although both fˆe2 (e2 ) and fˆe3 (e3 ) are skewed, fˆη|e (η|e) is fairly symmetric and centered around its mode for various values of e. Along with Figures 1.B.2 and 1.B.3, which illustrate density estimates of e and conditional density estimates of η given e for 2020, it supports the validity of using the mode as the estimate of η. 31 Figure 1.5.1: fˆe2 (e2 ) and fˆη2 |e2 (η2 |e2 ) for 2019 Figure 1.5.2: fˆe3 (e3 ) and fˆη3 |e3 (η3 |e3 ) for 2019 32 Table 1.5.5 shows the standard deviations of ξi2 and ξi3 , σ̂ξ2 and σ̂ξ3 , using the estimation results of the second step; recall that ξi2 and ξi3 represent the degree of allocative ineciency for the input pair (j, 1), j = 2, 3. Compared to the model estimates, the standard deviations of estimates for individual rm's allocative ineciency tend to be rather large but not too dierent. In addition, as the estimates of rm-level allocative ineciency, ξˆi2 and ξˆi3 , can be obtained, the results can be compared by the bank's classication, such as the banks' size 23 and the charter class . The results show that the degree of allocative ineciency of larger banks, such as banks with assets greater than $1 billion or nationally chartered commercial banks, and of thrifts is more dispersed than that of other banks. Table 1.5.5: Standard Deviations of ξˆi2 and ξˆi3 2019 2020 σ̂ξ2 σ̂ξ3 σ̂ξ2 σ̂ξ3 Model Estimates 0.5941 0.6447 0.6299 0.7005 Full Sample 0.7693 0.7495 0.7313 0.7286 Asset Size - Large Banks 0.8805 0.7667 0.8416 0.7617 - Small Banks 0.7476 0.7459 0.7016 0.7176 Charter Class -N 0.8024 0.7422 0.7614 0.7185 - NM 0.7571 0.7461 0.7102 0.7232 - SM 0.7354 0.7060 0.7137 0.6975 - SB 0.7732 0.7475 0.7319 0.7040 - SA 0.8287 0.8488 0.8434 0.8465 23 Banks are classied by its asset size and classication codes assigned by the FDIC, which indicate an institution's charter type, an institution's charter type, its Federal Reserve membership status, and its primary federal regulator. Please refer to Table 1.B.3 for more details. 33 Figure 1.5.3 shows the distribution of estimates for cost increases due to allocative inef- ciency, uAi , of individual banks, which is obtained from the third step of the process. The distribution shape is fairly similar to the results in Kumbhakar and Tsionas (2005b) in that it is highly skewed to the right. However, magnitudes are smaller seemingly owing to dierent denitions of inputs and outputs, sample periods, and the assumption about the variation of the term representing allocative ineciency, ξi . Even though a small number of estimates are negative, contrary to the denition of uA i , their magnitudes are not considerable. Figure 1.5.3: Distribution of ûA i (a) 2019 (b) 2020 Table 1.5.6 shows the average cost increases due to allocative ineciency by the banks' classication. The estimates of larger banks, such as banks with assets greater than $1 billion or nationally chartered commercial banks, and of thrifts are higher than those of other banks. In addition, the average cost increases due to allocative ineciency has not been changed signicantly between 2019 and 2020 by banks' asset sizes and the charter classes. Table 1.5.6: Average of ûA i 2019 2020 Full Sample 0.0251 0.0244 Asset Size - Large Banks 0.0327 0.0332 - Small Banks 0.0237 0.0223 Charter Class -N 0.0288 0.0282 - NM 0.0240 0.0229 - SM 0.0239 0.0242 - SB 0.0225 0.0210 - SA 0.0316 0.0326 34 Figure 1.5.4 shows the distribution of estimates for cost increases due to technical in- eciency, uTi , of individual banks, which is obtained from the last step of the process. As opposed to the distribution of ûA i , it is slightly skewed to the right. In addition, the estimates for cost increases due to technical ineciency are less dispersed in 2020 compared to those of the previous year. Its standard deviation is decreased from 0.0567 in 2019 to 0.0452 in 2020; this implies that banks became somewhat homogeneous in terms of technical ineciency. Figure 1.5.4: Distribution of ûTi (a) 2019 (b) 2020 Table 1.5.7 shows the average cost increases due to technical ineciency by the banks' classication. The estimates are fairly similar to those of uTi obtained from the model. Also, unlike the results of the third step, the estimates of the large banks are slightly smaller than those of the small banks. By the banks' charter class, nationally chartered commercial banks seem to be technically less ecient than others, but the gap between them decreased in 2020. Table 1.5.7: Average of ûTi 2019 2020 Model Estimates 0.1789 0.1643 Full Sample 0.1748 0.1620 Asset Size - Large Banks 0.1737 0.1619 - Small Banks 0.1750 0.1621 Charter Class -N 0.1819 0.1666 - NM 0.1743 0.1610 - SM 0.1751 0.1634 - SB 0.1641 0.1569 - SA 0.1710 0.1626 35 Table 1.5.8 summarizes the estimation results of individual technical and allocative in- eciency. To sum up, rst, the ratio of cost increase due to allocative ineciency, uAi , has not changed between 2019 and 2020. However, the ratio of cost increase due to technical ineciency, uTi , has decreased. It suggests the possibility of changes in the cost frontier during the pandemic, while banks try to maintain resource allocation. Second, cost increase due to ineciency is generally higher for larger banks. It is in line with previous studies, such as Altunbas et al. (2007) and Ding and Sickles (2019). Table 1.5.8: Average of ûTi and ûA i 2019 2020 ûTi ûA i ûTi ûA i Full Sample 0.1748 0.0251 0.1620 0.0244 Asset Size - Large Banks 0.1737 0.0327 0.1619 0.0332 - Small Banks 0.1750 0.0237 0.1621 0.0223 Charter Class -N 0.1819 0.0288 0.1666 0.0282 - NM 0.1743 0.0240 0.1610 0.0229 - SM 0.1751 0.0239 0.1634 0.0242 - SB 0.1641 0.0225 0.1569 0.0210 - SA 0.1710 0.0316 0.1626 0.0326 1.6 Conclusion Cost eciency analysis has the virtue that it enables researchers to decompose ineciency into two main sources; input-oriented technical ineciency, and resource misallocation. How- ever, there is no satisfactory method to measure and decompose both types of ineciency when exible functional forms are allowed for. In this paper, a model and an estimation strategy for the translog cost system are devel- oped to overcome limitations of previous stochastic cost frontier studies. By employing APS copulas, one can model the dependence between technical and allocative ineciency as well as provide a solution to the Greene Problem. The model can be estimated by the method of simulated likelihood. The proposed estimation strategy is developed upon economic in- 36 tuition behind the stochastic frontier model. It is also uncomplicated, as random numbers can be drawn from the simple density, the standard uniform density. In addition, a strategy to estimate individual ineciency is proposed, which uses not only conditional distributions as in previous studies, but also density deconvolution. An empirical exercise for the U.S. banking industry in 2019 and 2020 shows that the costs of U.S. banks increased by around 20% during the sample period due to ineciency, where technical and allocative ineciency account for around 16∼18% and 2.5%, respectively. During the pandemic, banks' technical ineciency has slightly decreased seemingly due to changes in the composition of costs and output, while the degree of allocative ineciency has not changed signicantly. Lastly, the results suggest that it would produce less plausible results when ignoring the dependence between technical and allocative ineciency. 37 APPENDICES 38 APPENDIX A Additional Details 1.A.1 Proof of and PJ PJ j=1 sij (y i , wi ) = 1 j=1 ηij = 0 1.A.1.1 PJ j=1 sij (wi , y i ) = 1 X J sj (y i , wi ) j=1 X J  X J XM  yw = βjw + ww βjk (lnwik ) + βmj (lnyim ) j=1 k=1 m=1 X J X J X J X J X M yw = βjw + ww βjk (lnwik ) + βmj (lnyim ) j=1 j=1 k=1 j=1 m=1 X J X J X J X M X J ww yw = βjw + βjk (lnwik ) + βmj (lnyim ) j=1 k=1 j=1 m=1 j=1 X J X J X J X M X J ww yw = βjw + (lnwik ) βjk + (lnyim ) βmj j=1 k=1 j=1 m=1 j=1 = 1. PJ The last equality holds because of the restrictions on the parameters, such as j=1 βjw = 1, PJ ww PJ ww PJ yw k=1 βjk = j=1 βjk = 0 ∀j or ∀k , and j=1 βmj = 0 ∀m. 1.A.1.2 PJ j=1 ηij = 0 PJ PJ PJ Since j=1 sij = 1 and j=1 sij (y i , wi ) = 1, j=1 ηij = 0 is guaranteed. For J =2 and 3, it can be also shown using the formula of ηij . (i) J =2 ηi1 + ηi2 s1 (y i , wi )[1 − {s∗i1 + (s∗i2 /eξi2 )}] + β12 ww ξi2 = ∗ ∗ ξ {si1 + (si2 /e )} i2 s2 (y i , wi )[1 − {s∗i1 + (s∗i2 /eξi2 )}eξi2 ] + β22 ww ξi2 + ∗ ∗ ξ ξ {si1 + (si2 /e i2 )}e i2 39 s1 (y i , wi )eξi2 [1 − {s∗i1 + (s∗i2 /eξi2 )}] + β12 ww ξi2 eξi2 = {s∗i1 + (s∗i2 /eξi2 )}eξi2 s2 (y i , wi )[1 − {s∗i1 + (s∗i2 /eξi2 )}eξi2 ] + β22 ww ξi2 + ∗ ∗ ξ ξ {si1 + (si2 /e i2 )}e i2 ξi2 s1 (y i , wi )e + s2 (y i , wi ) = {s∗i1 + (s∗i2 /eξi2 )}eξi2 s1 (y i , wi ) + s2 (y i , wi ) {s∗i1 + (s∗i2 /eξi2 )}eξi2 + β12  ww ξi2 eξi2 + β22ww ξi2 − ∗ ∗ ξ ξ {si1 + (si2 /e i2 )}e i2 s1 (y i , wi ) − si1 + β12 ξi2 eξi2 + s2 (y i , wi ) − s∗i2 + β22 ∗ ww  ww  ξi2 = {s∗i1 + (s∗i2 /eξi2 )}eξi2 = 0. The fourth equality holds as s1 (y i , wi ) + s2 (y i , wi ) = 1, and the last equality holds by the denition of sij . (ii) J =3 ηi1 + ηi2 + ηi3 s1 (y i , wi )[1 − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}] + β12 ww ww ξi2 + β13 ξi3 = ∗ ∗ ξ ∗ ξ {si1 + (si2 /e i2 ) + (si3 /e i3 )} s2 (y i , wi )[1 − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 ] + β22 ww ξi2 + β23 ww ξi3 + ∗ ∗ ξ ∗ ξ ξ {si1 + (si2 /e ) + (si3 /e )}e i2 i3 i2 s3 (y i , wi )[1 − {si1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi3 ] + β32 ∗ ww ξi2 + β33 ww ξi3 + ∗ ∗ ξ ∗ ξ ξ {si1 + (si2 /e ) + (si3 /e )}e i2 i3 i3 s1 (y i , wi )e e [1 − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}] ξi2 ξi3 = {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 β ww ξi2 eξi2 eξi3 + β13 ww ξi3 eξi2 eξi3 + ∗ 12 ∗ ξi2 {si1 + (si2 /e ) + (s∗i3 /eξi3 )}eξi2 eξi3 s2 (y i , wi )eξi3 [1 − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 ] + {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 ww β22 ξi2 eξi3 + β23 ww ξi3 eξi3 + ∗ {si1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 s3 (y i , wi )eξi2 [1 − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi3 ] + {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 ww β32 ξi2 eξi2 + β33 ww ξi3 eξi2 + ∗ {si1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 40 s1 (y i , wi )eξi2 eξi3 + s2 (y i , wi )eξi3 + s3 (y i , wi )eξi2 = {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 s1 (y i , wi ) + s2 (y i , wi ) + s3 (y i , wi ) eξi2 eξi3 {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}  − {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 β ww ξi2 eξi2 eξi3 + β13 ww ξi3 eξi2 eξi3 + ∗ 12 ∗ ξi2 {si1 + (si2 /e ) + (s∗i3 /eξi3 )}eξi2 eξi3 ww β22 ξi2 eξi3 + β23ww ξi3 eξi3 + ∗ {si1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 ww β32 ξi2 eξi2 + β33ww ξi3 eξi2 + ∗ {si1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 eξi2 eξi3 s1 (y i , wi ) − s∗i1 + β12 ww ww  ξi2 + β13 ξi3 = {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 eξi3 s2 (y i , wi ) − s∗i2 + β22 ww ww  ξi2 + β23 ξi3 + {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 eξi2 s3 (y i , wi ) − s∗i3 + β32 ww ww  ξi2 + β33 ξi3 + {s∗i1 + (s∗i2 /eξi2 ) + (s∗i3 /eξi3 )}eξi2 eξi3 = 0. The fourth equality holds as s1 (y i , wi ) + s2 (y i , wi ) + s3 (y i , wi ) = 1, and the last equality holds by the denition of sij . 1.A.2 Simplifying uAi and ηij PJ ∗ ξk Assume that there is no additive noise term in the cost share equations. Let Gi = k=1 (sik /e ). Then, PJ sj (y i , wi )(1 − Gi eξj ) + k=1 ww βjk ξk ηij = Gi eξj XJ ⇒ Gi ηij eξj = sj (y i , wi )(1 − Gi eξj ) + ww βjk ξk k=1 XJ  ξj ww ⇒ Gi sj (y i , wi ) + ηij e = sj (y i , wi ) + βjk ξk k=1 XJ ⇒ Gi sij eξj = sj (y i , wi ) + ww βjk ξk k=1 41 X J X J  X J  ξj ww ⇒ Gi sij e = sj (y i , wi ) + βjk ξk j=1 j=1 k=1 X J X J X J X J ⇒ Gi sij eξj = sj (y i , wi ) + ww βjk ξk j=1 j=1 j=1 k=1 1 ∴ Gi = PJ j=1 sij eξj PJ PJ PJ ww The last equality holds because j=1 sj (y i , wi ) = 1 and j=1 k=1 βjk ξk = 0 by the symmetry and the linear homogeneity conditions of the cost function. Also, note that X J ξj ww Gi sij e = sj (y i , wi ) + βjk ξk k=1 XJ ξj ww ⇒ Gi sij e = sij − ηij + βjk ξk k=1 XJ ww ∴ ηij = sij (1 − Gi eξj ) + βjk ξk k=1 Therefore, uA i and ηij can be simplied as J J X J J J M X J X X 1 X X ww X yw uA i = βjw ξj + ww βjk (lnwij )ξk + β ξj ξk + βmj (lnyim )ξj j=1 j=1 k=1 2 j=1 k=1 jk m=1 j=1 X J  ξj − ln sij e j=1 J  eξj  X ww ηij = sij 1 − PJ + βjk ξk . s e ξk k=1 ik k=1 1.A.3 Rosenblatt Transformation This subsection is written based on Rosenblatt (1952), Chapter 6.9.1 of Joe (2014), and Appendix B of Melchers and Beck (2018). Rosenblatt (1952) shows that a dependent random vector X = (X1 , · · · , Xk ) may be transformed to the random vector Z = (Z1 , · · · , Zk ), where i.i.d Zi ∼ U [0, 1] ∀ i = 1, · · · , k . This subsection summarizes the procedure for generating a dependent random vector X from the independent random vector. 42 Let F1:k (χ1 , · · · , χk ) be a multivariate distribution of X with marginal distributions F1 , · · · , Fk , where the corresponding random variables X1 , · · · , Xk can be continuous, dis- crete, or mixed. Rosenblatt (1952) proposes the transformation TR such that ζ = (ζ1 , · · · , ζk ) = TR (χ1 , · · · , χk ), where ζ1 = P(X1 ≤ χ1 ) = F1 (χ1 ) ζ2 = P(X2 ≤ χ2 |X1 = χ1 ) = F2|1 (χ2 |χ1 ) ζ3 = P(X3 ≤ χ3 |X1 = χ1 , X2 = χ2 ) = F3|12 (χ3 |χ1 , χ2 ) . . . ζk = P(Xk ≤ χk |X1 = χ1 , · · · , Xk−1 = χk−1 ) = Fk|1,··· ,k−1 (χk |χ1 , · · · , χk−1 ). With all the conditional distributions F1 , F2|1 , · · · , Fk|1,··· ,k−1 and their inverse functions, one can successively obtain dependent random numbers (χ1 , · · · , χk ) from independent uniform random numbers (ζ1 , · · · , ζk ) on [0, 1]k such that χ1 = F1−1 (ζ1 ) −1 χ2 = F2|1 (ζ2 |χ1 ) −1 χ3 = F3|12 (ζ3 |χ1 , χ2 ) . . . −1 χk = Fk|1,··· ,k−1 (ζk |χ1 , · · · , χk−1 ). To sum up, the consecutive process to generate (χ1 , · · · , χk ) is as follows: (i) Derive conditional distributions F2|1 , F3|12 , · · · , Fk|1,··· ,k−1 (ii) Draw ζ1 from U [0, 1] and obtain χ1 from χ1 = F1−1 (ζ1 ) or by solving F1 (χ1 ) − ζ1 = 0 −1 (iii) Given χ1 , draw ζ2 from U [0, 1] and obtain χ2 from χ2 = F2|1 (ζ2 |χ1 ) or by solving F2|1 (χ2 |χ1 ) − ζ2 = 0 43 −1 (iv) Given χ1 and χ2 , draw ζ3 from U [0, 1] and obtain χ3 from χ3 = F3|12 (ζ3 |χ1 , χ2 ) or by solving F3|12 (χ3 |χ1 , χ2 ) − ζ3 = 0, and so on. Note that a permutation of the order 1, · · · , k is possible, so one would choose a permu- tation in practice where the computations are simplest. For example, one can choose the reverse order to obtain dependent random numbers (χ1 , · · · , χk ) from independent uniform random numbers (ζ1 , · · · , ζk ) on [0, 1]k such that −1 χ1 = F1|2,··· ,k (ζ1 |χ2 , · · · , χk ) . . . −1 χk−2 = Fk−2|k−1,k (ζk−2 |χk−1 , χk ) −1 χk−1 = Fk−1|k (ζk−1 |χk ). χk = Fk−1 (ζk ) 1.A.4 Density of ϵi = vi + ui Let ϵi = vi +ui , where vi ∼ N (0, σv2 ), ui ∼ |N (0, σu2 )|, and vi and ui are independent. Aigner   h  i T 2 ϵ̃ ϵ̃λ et al. (1977) derive the density of a random variable ϵ̃i = vi − ui as σ · ϕ σ · 1 − Φ σ , where σ 2 = σv2 + σu2 , λ = σu /σv , and ϕ and Φ are density and distribution functions of 24 the standard normal distribution. Similarly, the density of ϵi = vi + ui can be derived as follows. Note that the marginal densities of vi and ui are 1  v2  fv (v) = √ exp − 2 , v ∈ R σv 2π 2σv 2  u2  fu u = √ exp − 2 , v ∈ R+ . σu 2π 2σu Since vi and ui are independent, the joint density of them is 1 2  v2 u2  fv,u (v, u) = √ √ exp − 2 − 2 σv 2π σu 2π 2σv 2σu 2 1 2 n 1 v  u2 o = √ √ exp − + , σv 2π σu 2π 2 σv2 σu2 24 Please refer to Ch.11.7.3 of Sickles and Zelenyuk (2019) for more details. 44 and the joint density of ϵi and ui becomes fϵ,u (ϵ, u) = fv,u (ϵ − u, u) 1 2 n 1  (ϵ − u)2 u2  o = √ √ exp − + 2 . σv 2π σu 2π 2 σv2 σu | O {z 1 } | O{z 2 } O O 1 and 2 can be transformed as follows: O : σ σσ √+2πσ pσ +2σ √2π p 2 2 v u 1 2 2 v u v u O : ϵ − 2ϵu 2 2 σ 2 v +u 2 + u σ 2 u 2 p p ϵ2 (σv2 + σu2 ) (2ϵu/ σv2 + σu2 )( σv2 + σu2 /σv σu ) u2 (σv2 + σu2 ) = − + σv2 (σv2 + σu2 ) σv2 /(σv σu ) σv2 σu2 p 2 ϵ  σu2  2ϵu σv2 + σu2 σu σ2 + σ2 = 1+ 2 − p + u2 v 2 2 u . σv2 + σu2 σv σv2 + σu2 σv σu σv σv σu Let σ 2 = σv2 + σu2 , λ = σu /σv , and δ 2 = (σv2 + σu2 )/(σv2 σu2 ). Then, δ 2 n 1  ϵ2 ϵ2 λ2 ϵu o 2 2 fϵ,u (ϵ, u) = √ √ exp − + − 2 δλ + u δ 2π σ 2π 2 σ2 σ2 σ 2  2  ϵ δ n 1 ϵλ  2 o = √ exp − 2 √ exp − − uδ . σ 2π 2σ 2π 2 σ As the support of u is [0, ∞), the density of ϵi is Z ∞ fϵ (ϵ) = fϵ,u (ϵ, u)du 0 ∞ ϵ2  Z 2  δ n 1  ϵλ 2 o = √ exp − 2 √ exp − − uδ du. σ 2π 2σ 0 2π 2 σ ϵλ ϵλ dγ Let γ= σ − uδ . Then, if u = 0, γ = σ , and if u → ∞, γ → −∞. Also, du = −δ , that is du = − 1δ dγ . Therefore, 2 1 n 1  ϵ 2 o Z −∞ δ  1  1 fϵ (ϵ) = √ exp − √ exp − γ 2 − dγ σ 2π 2 σ ϵλ σ 2π 2 δ 2 ϵ n −∞ o = ·ϕ · − Φ(γ) ϵλ σ σ σ 2  ϵ   ϵλ  = ·ϕ ·Φ . σ σ σ 45 APPENDIX B Tables and Figures Table 1.B.1: Descriptive Statistics of Key Variables for 2019 Mean Std. Dev Min Max Total Cost (Ci , $ million) 82.7 1,211.5 0.1 51,615.0 Interest Expenses (Ci1 , $ million) 30.3 434.2 0.0 17,008.0 Salaries (Ci2 , $ million) 43.5 662.2 0.1 28,538.0 Fixed Assets Expenses (Ci3 , $ million) 8.9 140.6 0.0 6,069.0 Loans (yi1 , $ million) 1,995.6 26,598.9 0.0 969,383.0 Other Earning Assets (yi2 , $ million) 1,274.8 24,618.5 0.3 1,174,359.0 Funds that Bank owes (xi1 , $ million) 3,054.9 47,236.1 0.5 1,980,733.1 Number of Full-time Employees (xi2 ) 402 5,552 2 232,982 Fixed Assets (xi3 , $ million) 36.6 504.6 0.0 22,432.0 wi1 = Ci1 /xi1 0.0090 0.0045 0.0000 0.0381 wi2 = Ci2 /xi2 0.0839 0.0283 0.0056 0.3706 wi3 = Ci3 /xi3 0.4039 0.8936 0.0075 22.7273 Table 1.B.2: Descriptive Statistics of Key Variables for 2020 Mean Std. Dev Min Max Total Cost (Ci , $ million) 73.0 1,000.1 0.0 40,331.0 Interest Expenses (Ci1 , $ million) 15.3 172.6 0.0 7,459.0 Salaries (Ci2 , $ million) 47.9 706.8 0.0 28,982.0 Fixed Assets Expenses (Ci3 , $ million) 9.8 151.8 0.0 6,362.0 Loans (yi1 , $ million) 2,111.3 26,425.0 0.0 995,415.0 Other Earning Assets (yi2 , $ million) 1,872.9 36,520.7 0.3 1,801,495.0 Funds that Bank owes (xi1 , $ million) 3,737.5 58,393.4 0.5 2,626,377.0 Number of Full-time Employees (xi2 ) 417 5,689 2 233,403 Fixed Assets (xi3 , $ million) 38.1 523.5 0.0 23,184.0 wi1 = Ci1 /xi1 0.0062 0.0035 0.0000 0.0285 wi2 = Ci2 /xi2 0.0889 0.0306 0.0000 0.3616 wi3 = Ci3 /xi3 0.4005 0.7763 0.0025 13.0000 Table 1.B.3: Classication of Banks Asset Size - Large Banks Banks with assets greater than $1 billion - Small Banks Banks with assets less than $1 billion Charter Class -N Commercial banks, federal charter, fed member - NM Commercial banks, state charter, fed non-member - SM Commercial or savings banks, state charter, fed member - SB Savings banks, state charter - SA Thrifts, federal or state charter 46 Figure 1.B.1: Sample Correlations (The APS-3-A Copula) (a) corr(ω1 , ω2 ) (b) corr(ω1 , (ω2 − 0.5)2 ) (c) corr(ω1 , ω3 ) (d) corr(ω1 , (ω3 − 0.5)2 ) 47 Figure 1.B.2: fˆe2 (e2 ) and fˆη2 |e2 (η2 |e2 ) for 2020 Figure 1.B.3: fˆe3 (e3 ) and fˆη3 |e3 (η3 |e3 ) for 2020 48 CHAPTER 2 MEASUREMENT AND DECOMPOSITION OF COST INEFFICIENCY USING COPULAS: EVIDENCE FROM MONTE CARLO SIMULATIONS 2.1 Introduction This paper provides methods for copula-based simulations and demonstrates the performance of the estimation strategy proposed by Ryu (2021). First, a method to generate pseudo data using the APS copulas developed by Amsler et al. (2021) is presented; it applies the inverse Rosenblatt transformation and the inverse transformation method. Second, given the data generating process, quasi-Monte Carlo simulations are conducted in order to conrm the validity of the estimation strategy in Ryu (2021) that can measure and decompose technical and allocative ineciency. Amsler et al. (2021) conduct Monte Carlo simulations to show that the stochastic pro- duction frontier model employing the APS-2-A copula can be reliably estimated. The model used for their simulations can be written as yi = α + β1 xi1 + β2 xi2 + vi − uTi   β1 wi2 xi1 − xi2 = ln + ei2 , (2.1.1) β2 wi1 where yi is the natural log of output of producer i, xi1 and xi2 are the natural log of inputs, vi ∈ R is a random disturbance, uTi ∈ R+ represents technical ineciency, wij ∈ R++ , j = 1, 2, are the price of input j , and ei2 ∈ R is a two-sided term capturing allocative ineciency and noise. They assume that a rm produces one output given two inputs, and uTi and ei2 are linked by the APS-2 copulas such that uTi is uncorrelated with ei2 but positively correlated with |ei2 |. Since their model uses the method of simulated likelihood for estimation, it requires to draw a N ×1 vector of random numbers, where N is the number of producers, from the distribution of uTi , such as the half normal distribution. This is because the joint 49 density of their model is fϵ,e2 (ϵ, e2 ) = fe2 (e2 ) · EuT [c(ω1 , ω2 ) · fv (ϵ + uT )], where ϵ = v − uT , EuT represents the expectation with respect to the distribution of uTi , ω1 = F1 (uT ), ω2 = F2 (e2 ), and F1 (uT ) and F2 (e2 ) are marginal cumulative distribution functions of uTi and ei2 , respectively. Furthermore, if a rm produces one output using three inputs, we can extend the above model, and the joint density in this case becomes fϵ,e2 ,e3 (ϵ, e2 , e3 ) = fe2 (e2 ) · fe3 (e3 ) · EuT [c(ω1 , ω2 , ω3 ) · fv (ϵ + uT )], where ω3 = F3 (e3 ), and F3 (e3 ) is a marginal cumulative distribution function of ei3 (Amsler et al., 2021, p.6). It implies that it requires to draw a N ×1 vector of random numbers as well even if the number of inputs increases. However, numerous studies on eciency analysis have assumed that producers use more than two inputs. Furthermore, we need to conrm whether an estimation strategy employing copulas would produce reliable estimates in more complex settings. For example, consider a translog cost system that can measure and decompose technical and allocative ineciency, which can be written as lnCi = lnC(y i , wi ) + vi + uTi + g(ξ i ) sij = sj (y i , wi ) + hj (ξ i ) + νij , j = 2, · · · , J, (2.1.2) where Ci is the actual cost of producer i, C(y i , wi ) is the deterministic kernel of the stochastic cost frontier, y i ∈ RM + is a vector of M outputs, wi ∈ RJ++ is a vector of input prices, vi ∈ R is a random disturbance, uTi ∈ R+ represents a cost increase due to technical ineciency, g(ξ i ) ∈ R+ represents a cost increase due to allocative ineciency, ξ i = (ξi2 , · · · , ξiJ ), ξij represents producers' allocative ineciency for the input pair (j, 1), sij ∈ [0, 1] is the actual cost share of input j , sj (y i , wi ) ∈ [0, 1] is the optimum cost share of input j , hj (ξ i ) ∈ R is the disturbance due to allocative ineciency, and νij ∈ R is additive noise. We can assume 50 that uTi and ξi are linked by the APS copulas such that uTi is uncorrelated with ξi2 , · · · , ξiJ but positively correlated with |ξi2 |, · · · , |ξiJ |. Then, the joint density of this model is       fϵ,ν (ϵ, ν) = Eζ fv ϵ − z1 T (ζ) − g z 2 T (ζ) ; θ       · fν e2 − h2 z 2 T (ζ) , · · · , eJ − hJ z 2 T (ζ) ; θ , where Eζ represents the expectation with respect to the distribution of ζ = (ζ1 , · · · , ζJ ), ζj ∼ U [0, 1], ϵ = v + uT + g(ξ), z1 and z2 are functions that transform CDF values to random numbers, T is the inverse function of the Rosenblatt transformation, and θ is the parameters. Contrary to Amsler et al. (2021), it requires to draw N ×J uniform random numbers ζ, where J represents the number of inputs, that each columns are uncorrelated. Hence, it is necessary to conduct another set of simulations that allows more inputs in order to check the validity of the economic model applying the APS copulas. In addition, methods of data generating process need to be provided in order to conduct simulations. Lastly, the plausibility of the assumption in Greene (1980) can be examined by conducting simulations. As discussed in Bauer (1990) and Kumbhakar and Lovell (2000), an econometric issue occurs in a cost system that employs exible functional forms, such as a translog function. The key question is how to model the relationship between ui = uTi + g(ξ i ) and eij = hj (ξ i )+νij of (2.1.2). Greene (1980) proposes a solution to this problem, which assumes that the disturbance in the cost function, ϵi = vi + ui , and the disturbance in the cost share equations, eij , are independent. By conducting simulations using a pseudo-data set that assumes technical and allocative ineciency are linked by the APS copulas, we can verify the validity of the assumption in Greene (1980). The remainder of the chapter is organized as follows. Section 2.2 illustrates how to draw 1 observations from the APS-3-A copula that corresponds to a three-input case. Section 2.3 shows the data generating process. Section 2.4 presents the Monte Carlo simulation results. Section 2.5 concludes the chapter. 1 Amsler et al. (2021) provide the procedure to draw observations from the APS-2-A copula. It is summarized in Appendix A. 51 2.2 Simulating from the APS-3-A Copula Since a copula itself is a cumulative distribution function whose marginal distributions follow U [0, 1], one needs to nd the conditional copula in order to employ the Rosenblatt transfor- mation. Copula arguments can then be obtained by applying the process described in this section that focuses on the APS-3-A copula. 2.2.1 Derivation of Conditional Distributions Let c123 (ω1 , ω2 , ω3 ) be a copula density, where (ω1 , ω2 , ω3 ) ∈ [0, 1]3 are copula arguments. It is assumed that ω1 is uncorrelated with ω2 and ω3 but correlated with |ω2 −0.5| and |ω3 −0.5|. In addition, the dependence between ω2 and ω3 is captured by any bivariate copula. For computational ease, ω3 , ω2 , and ω1 are generated sequentially; this the use of the conditional copulas C1|23 (ω1 |ω2 , ω3 ) and C2|3 (ω2 |ω3 ) for the Rosenblatt transformation. In this subsection, the conditional copulas, C1|23 (ω1 |ω2 , ω3 ) and C2|3 (ω2 |ω3 ), are derived, then methods to obtain the copula arguments, ω3 , ω2 , and ω1 , are presented. 2.2.1.1 Conditional Copula C1|23 (ω1 |ω2 , ω3 ) Assume that ω2 and ω3 follow the bivariate normal copula such that C23 (ω2 , ω3 ) = Φ2 (Φ−1 (ω2 ), Φ−1 (ω3 ); ρ) ∂2 c23 (ω2 , ω3 ) = C23 (ω2 , ω3 ) ∂ω2 ∂ω3 ∂ 2 Φ2 (Φ−1 (ω2 ), Φ−1 (ω3 ); ρ) ∂Φ−1 (ω2 ) ∂Φ−1 (ω3 ) = ∂Φ−1 (ω2 )∂Φ−1 (ω3 ) ∂ω2 ∂ω3 −1 −1 ϕ2 (Φ (ω2 ), Φ (ω3 ); ρ) = ϕ(Φ−1 (ω2 ))ϕ(Φ−1 (ω3 ))   Φ−1 (ω2 )2 −2ρΦ−1 (ω2 )Φ−1 (ω3 )+Φ−1 (ω3 )2 √1 exp − 2(1−ρ2 ) 2π 1−ρ2 =     Φ−1 (ω2 )2 Φ−1 (ω3 )2 √1 exp − √1 exp − 2π 2 2π 2 1 h ρ Φ (ω2 ) − 2ρΦ (ω2 )Φ−1 (ω3 ) + ρ2 Φ−1 (ω3 )2 i 2 −1 2 −1 = p exp − , 1 − ρ2 2(1 − ρ2 ) 52 where Φ is the cumulative distribution function of the standard normal distribution, ϕ is the probability density function of the standard normal distribution, Φ2 is the cumulative distribution function of the standardized bivariate normal distribution, ϕ2 is the probability density function of the standardized bivariate normal distribution, and ρ is the correlation parameter. R Given that C1|23 (ω1 |ω2 , ω3 ) = c1|23 (t1 |ω2 , ω3 )dt1 , it is required to obtain the conditional density c1|23 (ω1 |ω2 , ω3 ), which is c123 (ω1 , ω2 , ω3 ) c1|23 (ω1 |ω2 , ω3 ) = f23 (ω2 , ω3 ) c123 (ω1 , ω2 , ω3 ) = c23 (F2 (ω2 ), F3 (ω3 ))f2 (ω2 )f3 (ω3 ) c123 (ω1 , ω2 , ω3 ) = . c23 (ω2 , ω3 ) The second inequality holds because  FXY (x, y) = CXY FX (x), FY (y) ∂ 2 FXY (x, y) ⇒ fXY (x, y) = ∂x∂y 2  ∂ CXY FX (x), FY (y) = ∂x∂y 2  ∂ CXY FX (x), FY (y) ∂FX (x) ∂FY (y) = ∂FX (x)∂FY (y) ∂x ∂y  = cXY FX (x), FY (y) fX (x)fY (y) and the third equality holds as ω2 , ω3 ∼ U [0, 1]. Note that c123 (ω1 , ω2 , ω3 ) = 1 + [1 + θ12 (1 − 2ω1 ){1 − 12(ω2 − 0.5)2 )} − 1] + [1 + θ13 (1 − 2ω1 ){1 − 12(ω3 − 0.5)2 )} − 1] + {c23 (ω2 , ω3 ) − 1} = g2 (1 − 2ω1 ) + g3 (1 − 2ω1 ) + c23 (ω2 , ω3 ) = h(1 − 2ω1 ) + c23 (ω2 , ω3 ), where g2 = θ12 {1 − 12(ω2 − 0.5)2 }, g3 = θ13 {1 − 12(ω3 − 0.5)2 }, and h = g2 + g3 . 53 Hence, the conditional copula C1|23 (ω1 |ω2 , ω3 ) is Z ω1 C1|23 (ω1 |ω2 , ω3 ) = c1|23 (t1 |ω2 , ω3 )dt1 0 ω1 h(1 − 2t1 ) + c23 (ω2 , ω3 ) Z = dt1 0 c23 (ω2 , ω3 ) 1 ω1 =[ {h(t1 − t21 ) + c23 (ω2 , ω3 )t1 }] c23 (ω2 , ω3 ) 0 1 = {h(ω1 − ω12 ) + c23 (ω2 , ω3 )ω1 }. c23 (ω2 , ω3 ) 2.2.1.2 Conditional Copula C2|3 (ω2 |ω3 ) R Given that C2|3 (ω2 |ω3 ) = c2|3 (t2 |ω3 )dt2 , it is required to obtain the conditional density c2|3 (ω2 |ω3 ), which is c23 (ω2 , ω3 ) c2|3 (ω2 |ω3 ) = = c23 (ω2 , ω3 ). f3 (ω3 ) The second equality holds because ω3 ∼ U [0, 1]. Hence, the conditional copula C2|3 (ω2 |ω3 ) is Z ω2 C2|3 (ω2 |ω3 ) = c2|3 (t2 |ω3 )dt2 0 ω2 ρ2 Φ−1 (t2 )2 − 2ρΦ−1 (t2 )Φ−1 (ω3 ) + ρ2 Φ−1 (ω3 )2 i Z 1 h = p exp − dt2 0 1 − ρ2 2(1 − ρ2 ) Let y2 = Φ−1 (t2 ) and y3 = Φ−1 (ω3 ). Note that dt2 = ϕ(y2 )dy2 . Then, integrate by substitu- tion such as Z Φ−1 (ω2 ) h ρ2 y 2 − 2ρy y + ρ2 y 2 i 1 2 2 3 3 C2|3 (ω2 |ω3 ) = p exp − 2 ϕ(y2 )dy2 −∞ 1−ρ 2 2(1 − ρ ) h i y 2 −2ρy2 y3 +ρ2 y32 Z Φ−1 (ω2 ) √1 exp − 2 1 2π 2(1−ρ2 ) = p h i ϕ(y2 )dy2 1 − ρ2 y2 −∞ √1 exp − 2 2π 2 Z Φ−1 (ω2 ) h (y − ρy )2 i 1 2 3 = √ p exp − dy2 2 2(1 − ρ 2) −∞ 2π 1 − ρ  Φ−1 (ω ) − ρΦ−1 (ω )  2 3 =Φ p . 1−ρ 2 54 2.2.2 Obtain Copula Arguments 2.2.2.1 Obtain ω3 As ω3 ∼ U [0, 1], ζ3 = F3 (ω3 ) = ω3 . Therefore, draw ζ3 from U [0, 1] and dene ω3 = ζ3 . 2.2.2.2 Obtain ω2 First, draw ζ2 from U [0, 1]. Then, one can obtain ω2 by solving the equation C2|3 (ω2 |ω3 )−ζ2 = 0. It yields  Φ−1 (ω ) − ρΦ−1 (ω )  2 3 Φ p − ζ2 = 0 1−ρ 2 Φ−1 (ω2 ) − ρΦ−1 (ω3 ) ⇒ p = Φ−1 (ζ2 ) 1−ρ 2 p ⇒ Φ−1 (ω2 ) = ρΦ−1 (ω3 ) + 1 − ρ2 Φ−1 (ζ2 ) p ⇒ ω2 = Φ ρΦ−1 (ω3 ) + 1 − ρ2 Φ−1 (ζ2 )  2.2.2.3 Obtain ω1 First, draw ζ1 from U [0, 1]. Then, one can obtain ω1 by solving the equation C1|23 (ω1 |ω2 , ω3 )− ζ1 = 0. It yields, 1 {(h(ω1 − ω12 ) + c23 ω1 } − ζ1 = 0 c23 ⇒ h(ω1 − ω12 ) + c23 ω1 − c23 ζ1 = 0 ⇒ hω12 − (h + c23 )ω1 + c23 ζ1 = 0 p (h + c23 ) ± (h + c23 )2 − 4hc23 ζ1 ⇒ ω1 = , 2h where c23 = c2|3 (ω2 |ω3 ). It is necessary to check whether the square roots are real numbers. Given that (ω2 , ω3 ) ∈ [0, 1]2 and (θ12 , θ13 ) ∈ [−0.5, 0.5]2 , one can nd the upper and lower bounds of g2 , g3 . If x ∈ [0, 1], 1 − 12(x − 0.5)2 ∈ [−2, 1]. Therefore, (g2 , g3 ) ∈ [−1, 1]2 and h ∈ [−2, 2]. Since 55 c23 > 0 and ζ1 ∈ [0, 1], (h + c23 )2 − 4hc23 ζ1 ≥ 0 when h ∈ [−2, 0). Also, for h ∈ [0, 2], (h + c23 )2 − 4hc23 ζ1 = (h − c23 )2 + 4hc23 (1 − ζ1 ) ≥ 0. Hence, the solutions are real numbers unless h = 0. The remained question is which solution to take. Rewrite the solutions as p (h + c23 ) ± (h + c23 )2 − 4hc23 ζ1 ω1 = p 2h p {(h + c23 ) ± (h + c23 )2 − 4hc23 ζ1 }{(h + c23 ) ∓ (h + c23 )2 − 4hc23 ζ1 } = p 2h{(h + c23 ) ∓ (h + c23 )2 − 4hc23 ζ1 } (h + c23 )2 − (h + c23 )2 + 4hc23 ζ1 = p 2h{(h + c23 ) ∓ (h + c23 )2 − 4hc23 ζ1 } 2c ζ = p 23 1 . (h + c23 ) ∓ (h + c23 )2 − 4hc23 ζ1 √ (h+c23 )+ (h+c23 )2 −4hc23 ζ1 Consider the rst solution ω1 = 2h = √2c23 ζ1 . Then, (h+c23 )− (h+c23 )2 −4hc23 ζ1 p the denominator (h + c23 ) − (h + c23 )2 − 4hc23 ζ1 < 0 for h ∈ [−2, 0), which violates the √ (h+c23 )− (h+c23 )2 −4hc23 ζ1 condition that ω1 ∈ [0, 1]. Now, consider the second solution ω1 = = 2h √2c23 ζ1 . Given that c23 > 0 and ζ1 ∈ [0, 1], it is required to show that (h+c23 )+ (h+c23 )2 −4hc23 ζ1 ω1 ∈ [0, 1]. It can be shown as follows: (i) ω1 ≥ 0 As c23 > 0 and ζ1 ∈ [0, 1], the numerator is positive. So it suces to show that the p denominator is strictly positive. Since (h + c23 )2 − 4hc23 ζ1 ≥ 0, the denominator is guaranteed to be strictly positive for h ∈ (−c23 , 2]; that is, if c23 > 2, the denominator is strictly positive. If c23 ∈ (0, 2] and h ∈ [−2, −c23 ), one needs to compare the values p of |h + c23 | and | (h + c23 )2 − 4hc23 ζ1 |, which is equivalent to compare their squares. Note that {(h + c23 )2 − 4hc23 ζ1 } − (h + c23 )2 = −4hc23 ζ1 > 0 unless ζ1 = 0. 2 Hence the denominator is strictly positive except the special case. (ii) ω1 ≤ 1 To nd the maximum value of ω1 given the arguments in the numerator, it is nec- essary to nd the minimum value of the denominator. Let λ(h) = (h + c23 ) + 2 As Z1 ∼ U [0, 1], P(Z1 = 0) = 0. 56 p (h + c23 )2 − 4hc23 ζ1 . Then, ∂λ(h) 1 1 = 1 + {(h + c23 )2 − 4hc23 ζ1 }− 2 {2(h + c23 ) − 4c23 ζ1 } ∂h 2 (h + c23 ) − 2c23 ζ1 =1+ p (h + c23 )2 − 4hc23 ζ1 ∂λ(h) If (h + c23 ) − 2c23 ζ1 ≥ 0, ∂h > 0. For (h + c23 ) − 2c23 ζ1 < 0, compare the values p of |(h + c23 ) − 2c23 ζ1 | and | (h + c23 )2 − 4hc23 ζ1 |, which is equivalent to compare the values of {(h + c23 ) − 2c23 ζ1 }2 and (h + c23 )2 − 4hc23 ζ1 . Note that {(h + c23 ) − 2c23 ζ1 }2 − (h + c23 )2 − 4hc23 ζ1 = (h + c23 )2 − 4c23 ζ1 (h + c23 ) − (h + c23 )2 + 4hc23 ζ1 = −4c23 ζ1 {(h + c23 ) − h} = −4c223 ζ1 ≤ 0, which implies √(h+c23 )−2c 2 23 ζ1 ≤ −1, so ∂λ(h) ∂h ≥ 0. Therefore, given c23 and ζ1 , λ(h) (h+c23 ) −4hc23 ζ1 √ (2−c23 )+ (−2+c23 )2 +8c23 ζ1 has the smallest value when h = −2. For h = −2, ω1 = 4 . Note that p (2 − c23 ) + (−2 + c23 )2 + 8c23 ζ1 ω1 = ≤1 4 p ⇔ (−2 + c23 )2 + 8c23 ζ1 ≤ 2 + c23 ⇔ (−2 + c23 )2 + 8c23 ζ1 ≤ (2 + c23 )2 ⇔ 4 − 4c23 + c223 + 8c23 ζ1 ≤ 4 + 4c23 + c223 ⇔ 8c23 (1 − ζ1 ) ≥ 0. The last inequality holds because of c23 > 0 and ζ1 ∈ [0, 1].■ Let A = h + c23 and B = c23 ζ1 , where c23 = c2|3 (ω2 |ω3 ). Then the solution is 2B ω1 = √ . A + A2 − 4hB 57 2.3 Generating Pseudo Data Consider the following stochastic cost frontier model such that lnCi = lnC(y i , wi ) + vi + uTi + g(ξ i ) sij = sj (y i , wi ) + hj (ξ i ) + νij , j = 2, · · · , J. Each component of the system dened in Section 2.1 can be written as M M M X y 1 X X yy lnC(y i , w i ) = β0 + βm (lnyim ) + β (lnyim )(lnyin ) m=1 2 m=1 n=1 mn J J J X 1 X X ww + βjw (lnwij ) + β (lnwij )(lnwik ) j=1 2 j=1 k=1 jk X M X J yw + βmj (lnyim )(lnwij ) m=1 j=1 XJ XM ww yw sj (y i , wi ) = βjw + βjk (lnwik ) + βmj (lnyim ), j = 2, · · · , J k=1 m=1 J J X J J J X X ww 1 X X ww g(ξ i ) = βjw ξij + βjk (lnwij )ξik + β ξij ξik j=1 j=1 k=1 2 j=1 k=1 jk X M XJ X J yw + βmj (lnyim )ξij + ln (s∗ij /eξij ) m=1 j=1 j=1 PJ ∗ ξik ξij PJ ww sj (y i , wi )[1 − { k=1 (sik /e )}e ] + k=1 βjk ξik hj (ξ i ) = PJ , j = 2, · · · , J, { k=1 (s∗ik /eξik )}eξij PJ where s∗ij = sj (y i , wi ) + k=1 ww βjk ξik , j = 2, · · · , J, is the shadow cost share of input j for producer i who is assumed to be only allocatively inecient. Four types of pseudo-data sets are generated by the pair of the numbers of inputs (J ) and outputs (M ): two inputs - one output, two inputs - two outputs, three inputs - one output, and three inputs - two outputs, where each set includes 1000 producers (N = 1000). In order to construct pseudo-data sets, it is necessary to draw three types of variables: (i) variables consisting of deterministic kernels of the translog cost system, yi and wi ; (ii) random components of the model not linked by the APS copulas, vi and νi ; and (iii) random components of the model linked by the APS copulas, uTi and ξi . The dependent variables 58 of the translog cost system, Ci and si2 , · · · , siJ , can be calculated using those variables and parameters. The rst and second types of variables are generated as follows. Note that the underlying production technology of the translog cost function is unknown, as the translog cost function has neither a closed-form dual production nor a transformation function (Kumbhakar and Lovell, 2000, p.154). Hence, outputs, lnyi1 and lnyi2 , are drawn from Γ(2, 2) and Γ(3, 2) with a correlation coecient of 0.8. Input prices, lnwi1 , lnwi2 , and lnwi3 , are independently drawn from N (1, 0.12 ), N (2, 0.12 ), and N (3, 0.12 ) distributions, respectively. Stochastic noise terms, vi and νi , are generated from N (0, σv2 ) and N (0, Σν ). The third type of variables are generated in the reverse order of the procedure for a change of variables that is used to derive the joint density in Ryu (2021). Figure 2.3.1 illustrates the procedure to simulate Z = (uTi , ξi2 , · · · , ξiJ ). Figure 2.3.1: Procedure to Simulate Z = (uTi , ξi2 , · · · , ξiJ ) To be specic, random components of the model linked by the APS copulas are simu- lated by the following procedure. First, draw independent random numbers, ζ, from the uniform distribution over [0, 1]. Second, produce CDF values, ω, given θ2 and the inverse of 59 3 conditional APS copula functions employing the inverse Rosenblatt transformation . Third, generate Z = (Z1 , Z 2 ) = (uTi , ξi2 , · · · , ξiJ ) given θ 3 and the inverse of cumulative distribution functions by the inverse transformation method. There is a practical issue in the rst step. In order to apply the inverse Rosenblatt transformation, it is essential to draw an N ×J array of uniformly distributed random numbers in which columns are uncorrelated in order to correctly estimate θ2 . Because of the 4 practical diculties to use truly random variables in the Monte Carlo methods , two methods for generating random numbers are generally used in applications: (i) pseudo-random number generators (PRNGs); and (ii) quasi-random number generators (QRNGs). Two functions provided by Matlab are considered to draw ζ1 , · · · , ζJ : (i) rand that generates uniformly distributed pseudo-random numbers; and (ii) haltonset that produces Halton sequences that make up the representative example of quasi-random number sequences. For illustrative purposes, Figure 2.B.1 shows sample correlations between two uniform random variables, ζ1 and ζ2 to compare the performance of the two functions. The number of replications is 1000, where three sample sizes N ∈ {100, 1000, 10000} are considered for each replication. Figures 2.B.1(a), 2.B.1(c), and 2.B.1(e) are obtained by the function rand. Figures 2.B.1(b), 2.B.1(d), and 2.B.1(f ) are obtained by the function haltonset, where several methods are applied to address the inherent issue that the points of a quasi-random 5 sequence are correlated. As shown in Figure 2.B.1, although the columns of the arrays produced by rand are theoretically uncorrelated, some pairs of ζ1 and ζ2 are highly correlated, especially when the number of draws are not suciently large. Also, even if N = 10000, some pairs of ζ1 and ζ2 seems to be signicantly correlated. By contrast, correlations between ζ1 and ζ2 generated 3 It is known as conditional distribution method. Please refer to Embrechts et al. (2003) and Cambou et al. (2017) for more details. 4 Please see Ch.8 of Judd (1998) for more details. 5 Matlab provides three methods: (i) omit initial points in the sequence; (ii) set interval between points; and (iii) scramble the sequence. In the simulation, the rst 100,000 values of the Halton point set are omitted, the every 100,001st point are retained, and then the Halton point set is scrambled by a reverse-radix operation. 60 by haltonset are mostly negligible. Hence, a QRNG, haltonset of Matlab , is used to generate ζ1 , · · · , ζJ . In the second step, ω1 , · · · , ωJ are generated through a consecutive process based on the Rosenblatt transformation by solving equations such that ζJ = FJ (ωJ ) ζJ−1 = FJ−1|J (ωJ−1 |ωJ ) ζJ−2 = FJ−2|J−1,J (ωJ−2 |ωJ−1 , ωJ ) . . . ζ1 = F1|2,··· ,J (ω1 |ω2 , · · · , ωJ ). As derived in Appendix A, ω1 and ω2 for the APS-2-A copula are generated as follows: ω2 = ζ2 2ζ1 ω1 = p , A + A2 − 4(A − 1)ζ1 where A = 1 + g2 and g2 = θ12 {1 − 12(ω2 − 0.5)2 }. For the association parameter of the APS-2-A copula, three values of θ12 ∈ {0, 0.2, 0.4} are considered. Also, Section 2.2 shows that ω1 , ω2 , and ω3 for the APS-3-A copula are generated as follows: ω3 = ζ3 p ω2 = Φ ρΦ−1 (ω3 ) + 1 − ρ2 Φ−1 (ζ2 )  2B ω1 = √ , A+ A2 − 4hB where ρ is the correlation parameter of the bivariate Gaussian copula, A = h + c23 (ω2 , ω3 ), h = g2 +g3 , g2 = θ12 {1−12(ω2 −0.5)2 }, g3 = θ13 {1−12(ω3 −0.5)2 }, and B = c23 (ω2 , ω3 )ζ1 . For the association parameter of the APS-3-A copula, two pairs of (θ12 , θ13 ) ∈ {(0, 0), (0.2, 0.2)} are considered to generate data sets, and the correlation parameter of the bivariate Gaussian copula ρ is set to −0.5. 61 2.4 Results of Monte Carlo Simulations Here, two sets of simulations are conducted. The rst set of simulations (Simulation I) is conducted to conrm the validity of the estimation strategy in Ryu (2021). The purpose of the second set of simulations (Simulation II) is to examine the plausibility of the assump- tion in Greene (1980) when technical and allocative ineciency are indeed dependent. The number of replications is 1,000 for both sets of simulations. 2.4.1 Simulation I A set of quasi-Monte Carlo simulations based on quasi-random sequences is conducted. Hal- ton sequences are also used for the simulations as the data generating process but for dierent reasons. The joint density of X and Y involves a multidimensional integral, in which a quasi- Monte Carlo integration is generally superior to standard Monte Carlo methods in terms of 6 integration error and its convergence rate. For example, Moroko and Caisch (1995) show that a quasi-Monte Carlo method using a Halton sequence has the lowest integration error and the fastest convergence rate up to around six dimensions among (quasi-)Monte Carlo methods using Halton, Sobol, Faure, and pseudo-random sequences. For copula sampling, in addition, Cambou et al. (2017) show that replacing PRNGs with QRNGs for integration also improves performance, reducing the variance of the obtained estimators and improving the convergence rate of the variance. Tables 2.4.1 to 2.4.4 report the results of quasi-Monte Carlo simulations for θ12 = 0.4 for two-input cases or θ12 = θ13 = 0.2 for three-input cases. Other results are reported in Tables 2.B.1 to 2.B.6 of Appendix B. R sets of N ×J Halton sequences are drawn for estimation, where R = 10, 000, N = 1, 000, and J =2 or 3. Although the standard deviations of the association parameters θ12 and θ13 are somewhat large, the results suggest that the stochastic cost frontier model in Ryu (2021) can be also reliably estimated like the stochastic production frontier model as in Amsler et al. (2021). That is, the modied translog cost system based on 6 Please refer to Caisch (1998) for more details. 62 Kumbhakar (1997) and the APS copulas is estimable by the maximum simulated likelihood estimator established on the probability integral transformation and the copula-based version of the Rosenblatt transformation. Table 2.4.1: Result of Simulation I (J = 2, M = 1, θ12 = 0.4) ¯ ¯ ¯ θ θTrue θ̂MSL θ θTrue θ̂MSL θ θTrue θ̂MSL β0 10.0000 9.9999 β1w 0.6000 0.6006 σv 0.3162 0.3154 (0.0306) (0.0127) (0.0071) β1y 0.8250 0.8249 ww β11 0.0500 0.0507 σν2 0.0100 0.0062 (0.0111) (0.0129) (0.0085) yy β11 0.0500 0.0500 yw β11 0.0100 0.0100 θ12 0.4000 0.3765 (0.0018) (0.0010) (0.1219) σT 0.2236 0.2239 (0.0170) σξ2 0.3162 0.3183 (0.0246) Note: Standard deviations are in parentheses. Table 2.4.2: Result of Simulation I (J = 2, M = 2, θ12 = 0.4) ¯ ¯ ¯ θ θTrue θ̂MSL θ θTrue θ̂MSL θ θTrue θ̂MSL β0 10.0000 9.9984 yy β22 0.0400 0.0402 σv 0.3162 0.3146 (0.0353) (0.0031) (0.0072) β1y 0.4000 0.4002 β1w 0.6000 0.6000 σ ν2 0.0100 0.0099 (0.0140) (0.0057) (0.0104) β2y 0.3000 0.2995 ww β11 0.0500 0.0500 θ12 0.4000 0.4014 (0.0130) (0.0032) (0.1113) yy β11 0.0500 0.0502 yw β11 0.0100 0.0100 σT 0.2236 0.2261 (0.0036) (0.0017) (0.0173) β12 -0.0100 -0.0102 yy β21 -0.0150 -0.0150 yw σξ2 0.3162 0.3112 (0.0032) (0.0010) (0.0162) Note: Standard deviations are in parentheses. 63 Table 2.4.3: Result of Simulation I (J = 3, M = 1, θ12 = θ13 = 0.2) ¯ ¯ ¯ θ θTrue θ̂MSL θ θTrue θ̂MSL θ θTrue θ̂MSL β0 10.0000 9.9993 ww β22 0.0250 0.0250 θ12 0.2000 0.1741 (0.0310) (0.0041) (0.1110) β1y 0.8250 0.8247 yw β11 0.0100 0.0100 θ13 0.2000 0.1780 (0.0113) (0.0004) (0.1109) yy β11 0.0500 0.0500 β12 -0.0050 -0.0051 yw ρθ -0.5000 -0.4996 (0.0019) (0.0009) (0.0176) β1w 0.2500 0.2495 σv 0.3162 0.3155 σT 0.2236 0.2252 (0.0027) (0.0072) (0.0171) β2w 0.4000 0.4006 σν2 0.0100 0.0095 σξ2 0.3162 0.3170 (0.0048) (0.0025) (0.0121) ww β11 0.0350 0.0347 σν3 0.0100 0.0100 σξ3 0.3162 0.3167 (0.0014) (0.0008) (0.0047) ww β12 -0.0150 -0.0148 ρν 0.0000 -0.0409 (0.0019) (0.0796) Note: Standard deviations are in parentheses. Table 2.4.4: Result of Simulation I (J = 3, M = 2, θ12 = θ13 = 0.2) ¯ ¯ ¯ θ θTrue θ̂MSL θ θTrue θ̂MSL θ θTrue θ̂MSL β0 10.0000 10.0000 ww β11 0.0350 0.0348 σν2 0.0100 0.0095 (0.0348) (0.0015) (0.0024) β1y 0.4000 0.3994 ww β12 -0.0150 -0.0149 σν3 0.0100 0.0100 (0.0139) (0.0019) (0.0005) β2y 0.3000 0.3001 ww β22 0.0250 0.0250 ρν 0.0000 -0.0665 (0.0129) (0.0033) (0.1010) yy β11 0.0500 0.0501 yw β11 0.0100 0.0100 θ12 0.2000 0.1791 (0.0038) (0.0007) (0.1108) yy β12 -0.0100 -0.0100 yw β12 -0.0050 -0.0050 θ13 0.2000 0.1788 (0.0033) (0.0011) (0.1095) yy β22 0.0400 0.0400 yw β21 0.0150 0.0150 ρθ -0.5000 -0.5000 (0.0032) (0.0004) (0.0127) β1w 0.2500 0.2496 yw β22 -0.0050 -0.0050 σT 0.2236 0.2243 (0.0026) (0.0007) (0.0173) β2w 0.4000 0.4000 σv 0.3162 0.3151 σξ2 0.3162 0.3164 (0.0032) (0.0071) (0.0078) σξ3 0.3162 0.3165 (0.0041) Note: Standard deviations are in parentheses. 64 2.4.2 Simulation II I conduct another set of simulations using the pseudo-data set described in Section 2.3, which assumes that technical and allocative ineciency are linked by the APS copulas. The objective of this simulation is to check the validity of the assumption in Greene (1980) when technical and allocative ineciency are actually dependent. Consider a stochastic cost frontier model such as lnCi = lnC(y i , wi ) + ϵi = lnC(y i , wi ) + vi + ui sij = sj (y i , wi ) + eij , j = 2, · · · , J. (2.4.1) i.i.d Greene (1980) assumes that ϵi is independent of ei = (ei2 , ei3 ). I assume that vi ∼ N (0, σv2 ), i.i.d i.i.d ui ∼ |N (0, σu2 )|, and ei ∼ N (0, Σe ). Maximum likelihood estimation can be applied to esti- mate parameters of the model, where the joint density of ϵi , ei2 , ei3 is simply fϵ,e2 ,e3 (ϵ, e2 , e3 ) = fϵ (ϵ) · fe (e), as ϵi and ei are assumed to be independent. 7 Tables 2.4.5 to 2.4.8 show the result of simulations when θ12 = 0.4 for J = 2 and θ12 = θ12 = 0.2 for J = 3. The key nding is as follows. As ui = uTi + g(ξ i ), we do not know the true standard deviations of ui . 8 However, as both uTi and g(ξ i ) are positive, the value of ui is higher than that of uTi . It implies that the standard deviation of ui , σu , should be higher than the standard deviation of uTi , σuT , as the mean of a random variable from the half normal distribution is an increasing function in its standard deviation. However, the estimates of σu for all cases are less than the true value of σT = 0.2236. It suggests that if one ignores the relationship between technical and allocative ineciency when they are indeed dependent, estimates of a cost increase due to ineciency would be biased. 7 Given the assumption about the distribution of vi and ui , the probability density function of ϵi is 2 ϵ  ϵλ  fϵ (ϵ) = ·ϕ ·Φ , σ σ σ where σ 2 = σv2 + σu2 , λ = σu /σv , and ϕ and Φ are density and distribution functions of the standard normal distribution. 8 In addition, as eij = hj (ξ i ) + νij , j = 2, 3, we do not know the true standard deviations of eij . 65 Table 2.4.5: Result of Simulation II (J = 2, M = 1, θ12 = 0.4) ¯ ¯ ¯ θ θTrue θ̂ML θ θTrue θ̂ML θ θTrue θ̂ML β0 10.0000 10.0213 β1w 0.6000 0.5984 σv 0.3162 0.3164 (0.0971) (0.0076) (0.0249) β1y 0.8250 0.8288 ww β11 0.0500 0.0491 σu - 0.1924 (0.0112) (0.0098) (0.1122) β11 0.0500 yy 0.0496 β11 0.0100 0.0099 yw σe2 - 0.0604 (0.0019) (0.0010) (0.0004) Note: Standard deviations are in parentheses. Table 2.4.6: Result of Simulation II (J = 2, M = 2, θ12 = 0.4) ¯ ¯ ¯ θ θTrue θ̂ML θ θTrue θ̂ML θ θTrue θ̂ML β0 10.0000 10.0096 yy β22 0.0400 0.0395 σv 0.3162 0.3143 (0.1045) (0.0034) (0.0300) β1y 0.4000 0.4004 β1w 0.6000 0.5987 σu - 0.2030 (0.0147) (0.0050) (0.1161) β2y 0.3000 0.3048 ww β11 0.0500 0.0494 σe2 - 0.0624 (0.0156) (0.0026) (0.0012) β11 0.0500 yy 0.0496 yw β11 0.0100 0.0099 (0.0039) (0.0016) yy β12 -0.0100 -0.0099 yw β21 -0.0150 -0.0148 (0.0034) (0.0008) Note: Standard deviations are in parentheses. Table 2.4.7: Result of Simulation II (J = 3, M = 1, θ12 = θ13 = 0.2) ¯ ¯ ¯ θ θTrue θ̂ML θ θTrue θ̂ML θ θTrue θ̂ML β0 10.0000 10.0755 ww β11 0.0350 0.0335 σv 0.3162 0.3342 (0.0543) (0.0015) (0.0114) β1y 0.8250 0.8286 ww β12 -0.0150 -0.0148 σu - 0.1338 (0.0113) (0.0016) (0.0588) yy β11 0.0500 0.0499 ww β22 0.0250 0.0238 σe2 - 0.0931 (0.0019) (0.0032) (0.0004) β1w 0.2500 0.2425 yw β11 0.0100 0.0101 σe3 - 0.0914 (0.0029) (0.0004) (0.0004) β2w 0.4000 0.3965 β12 -0.0050 -0.0038 yw ρe - -0.5559 (0.0043) (0.0009) (0.0014) Note: Standard deviations are in parentheses. 66 Table 2.4.8: Result of Simulation II (J = 3, M = 2, θ12 = θ13 = 0.2) ¯ ¯ ¯ θ θTrue θ̂MSL θ θTrue θ̂MSL θ θTrue θ̂MSL β0 10.0000 10.0712 β2w 0.4000 0.4008 yw β22 -0.0050 -0.0041 (0.0545) (0.0044) (0.0007) β1y 0.4000 0.3963 ww β11 0.0350 0.0330 σv 0.3162 0.3328 (0.0140) (0.0016) (0.0111) β2y 0.3000 0.3070 ww β12 -0.0150 -0.0149 σu - 0.1369 (0.0130) (0.0018) (0.0551) yy β11 0.0500 0.0495 ww β22 0.0250 0.0224 σe2 - 0.0842 (0.0038) (0.0035) (0.0004) yy β12 -0.0100 -0.0097 yw β11 0.0100 0.0097 σe2 - 0.0796 (0.0033) (0.0008) (0.0004) yy β22 0.0400 0.0391 yw β12 -0.0050 -0.0064 ρe - -0.5208 (0.0032) (0.0013) (0.0023) β1w 0.2500 0.2418 yw β21 0.0150 0.0150 (0.0035) (0.0004) Note: Standard deviations are in parentheses. 2.5 Conclusion The estimation strategy proposed in Ryu (2021) involves multidimensional integral and two- step transformations. Therefore, it would be necessary to conduct a set of Monte Carlo simulations to conrm their validity. Like Amsler et al. (2021), the simulation results sug- gest that the parameters of the model in which APS copulas are employed can be reliably estimated in complex settings. In addition, I conduct another set of simulations to check the plausibility of assumptions in Greene (1980). Simulation results imply that it would lead biased estimates of ineciency to ignore the relationship between technical and allocative ineciency when they are indeed dependent. 67 APPENDICES 68 APPENDIX A Simulating from the APS-2-A Copula This section is written based on the supplemental material for Amsler et al. (2021). Let c12 (ω1 , ω2 ) be a copula density, where (ω1 , ω2 ) ∈ [0, 1]2 are copula arguments that are uncor- related, but ω1 is correlated with |ω2 − 0.5|. For computational ease, ω2 and ω1 are generated sequentially, which requires C1|2 (ω1 |ω2 ) is used for the Rosenblatt transformation. In this section, C1|2 (ω1 |ω2 ) is derived, then meth- ods to obtain ω2 and ω1 are presented. 2.A.1 Derivation of the Conditional Distribution R Given that C1|2 (ω1 |ω2 ) = c1|2 (t1 |ω2 )dt1 , it is required to obtain the conditional density c1|2 (ω1 |ω2 ), which is c12 (ω1 , ω2 ) c1|2 (ω1 |ω2 ) = = c12 (ω1 , ω2 ). f2 (ω2 ) The second equality holds because ω2 ∼ U [0, 1]. Hence, the conditional copula C1|2 (ω1 |ω2 ) is Z ω1 C1|2 (ω1 |ω2 ) = c1|2 (t1 |ω2 )dt1 0 Z ω1 = [1 + θ12 (1 − 2t1 ){1 − 12(ω2 − 0.5)2 )}]dt1 0 ω1 = [t1 + θ12 (t1 − t21 ){1 − 12(ω2 − 0.5)2 )}] 0 = ω1 + θ12 ω1 (1 − ω1 ){1 − 12(ω2 − 0.5)2 )} = ω1 + gω1 (1 − ω1 ), where g = θ12 {1 − 12(ω2 − 0.5)2 }. 2.A.2 Obtain Copula Arguments 2.A.2.1 Obtain ω2 As ω2 ∼ U [0, 1], ζ2 = F2 (ω2 ) = ω2 . Therefore, draw ζ2 from U [0, 1] and dene ω2 = ζ2 . 69 2.A.2.2 Obtain ω1 First, draw ζ1 from U [0, 1]. Then, one can obtain ω1 by solving the equation C1|2 (ω1 |ω2 )−ζ1 = 0. It yields ω1 + gω1 (1 − ω1 ) − ζ1 = 0 ⇒ gω12 − (1 + g)ω1 + ζ1 = 0 p (1 + g) ± (1 + g)2 − 4gζ1 ⇒ ω1 = . 2g √ (1+g)+ (1+g)2 −4gζ1 Due to the upper and lower bounds of ω1 , the solution ω1 = 2g is ruled out (Amsler et al., 2021). Let A = 1 + g . Then, p A − A2 − 4(A − 1)ζ1 ω1 = 2(A − 1) p p (A − A2 − 4(A − 1)ζ1 )(A + A2 − 4(A − 1)ζ1 ) = p 2(A − 1)(A + A2 − 4(A − 1)ζ1 ) A2 − A2 + 4(A − 1)ζ1 = p 2(A − 1)(A + A2 − 4(A − 1)ζ1 ) 2ζ1 = p 2 A + A − 4(A − 1)ζ1 70 APPENDIX B Tables and Figures Figure 2.B.1: Sample Correlations between ζ1 and ζ2 (a) PRNG - rand of Matlab (b) QRNG - haltonset of Matlab (N = 100) (N = 100) (c) PRNG - rand of Matlab (d) QRNG - haltonset of Matlab (N = 1000) (N = 1000) (e) PRNG - rand of Matlab (f ) QRNG - haltonset of Matlab (N = 10000) (N = 10000) 71 Table 2.B.1: Result of Simulation I (J = 2, M = 1, θ12 = 0) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ β0 10.0000 9.9993 β1w 0.6000 0.6002 σv 0.3162 0.3153 (0.0305) (0.0130) (0.0071) β1y 0.8250 0.8250 ww β11 0.0500 0.0505 σν2 0.0100 0.0074 (0.0111) (0.0133) (0.0109) yy yw β11 0.0500 0.0500 β11 0.0100 0.0101 θ12 0.0000 0.0052 (0.0018) (0.0010) (0.1679) σT 0.2236 0.2243 (0.0170) σξ2 0.3162 0.3146 (0.0329) Note: Standard deviations are in parentheses. Table 2.B.2: Result of Simulation I (J = 2, M = 1, θ12 = 0.2) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ β0 10.0000 9.9996 β1w 0.6000 0.6008 σv 0.3162 0.3153 (0.0304) (0.0127) (0.0071) β1y 0.8250 0.8250 ww β11 0.0500 0.0510 σν2 0.0100 0.0074 (0.0110) (0.0129) (0.0103) yy yw β11 0.0500 0.0500 β11 0.0100 0.0101 θ12 0.2000 0.2056 (0.0018) (0.0010) (0.1596) σT 0.2236 0.2243 (0.0170) σξ2 0.3162 0.3167 (0.0294) Note: Standard deviations are in parentheses. 72 Table 2.B.3: Result of Simulation I (J = 2, M = 2, θ12 = 0) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ yy β0 10.0000 9.9989 β22 0.0400 0.0401 σv 0.3162 0.3145 (0.0352) (0.0031) (0.0072) β1y 0.4000 0.4002 β1w 0.6000 0.5998 σν2 0.0100 0.0160 (0.0140) (0.0056) (0.0164) β2y 0.3000 0.2996 ww β11 0.0500 0.0499 θ12 0.0000 0.0610 (0.0130) (0.0030) (0.2049) yy yw β11 0.0500 0.0501 β11 0.0100 0.0100 σT 0.2236 0.2263 (0.0036) (0.0017) (0.0173) yy yw β12 -0.0100 -0.0101 β21 -0.0150 -0.0150 σξ2 0.3162 0.2934 (0.0032) (0.0008) (0.0475) Note: Standard deviations are in parentheses. Table 2.B.4: Result of Simulation I (J = 2, M = 2, θ12 = 0.2) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ yy β0 10.0000 9.9983 β22 0.0400 0.0402 σv 0.3162 0.3144 (0.0352) (0.0031) (0.0072) β1y 0.4000 0.4002 β1w 0.6000 0.5999 σν2 0.0100 0.0132 (0.0140) (0.0057) (0.0138) β2y 0.3000 0.2996 ww β11 0.0500 0.05000 θ12 0.2000 0.2525 (0.0130) (0.0031) (0.1654) yy yw β11 0.0500 0.0501 β11 0.0100 0.0100 σT 0.2236 0.2266 (0.0036) (0.0017) (0.0174) yy yw β12 -0.0100 -0.0101 β21 -0.0150 -0.0150 σξ2 0.3162 0.3144 (0.0032) (0.0008) (0.0072) Note: Standard deviations are in parentheses. 73 Table 2.B.5: Result of Simulation I (J = 3, M = 1, θ12 = θ13 = 0) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ ww β0 10.0000 9.9990 β22 0.0250 0.0251 θ12 0.0000 -0.0049 (0.0310) (0.0044) (0.1237) yw β1y 0.8250 0.8247 β11 0.0100 0.0100 θ13 0.0000 -0.0002 (0.0113) (0.0004) (0.1186) yy yw β11 0.0500 0.0500 β12 -0.0050 -0.0050 ρ -0.5000 -0.4971 (0.0019) (0.0010) (0.0339) β1w 0.2500 0.2495 σv 0.3162 0.3154 σT 0.2236 0.2252 (0.0029) (0.0072) (0.0171) β2w 0.4000 0.4005 σν2 0.0100 0.0094 σξ2 0.3162 0.3177 (0.0060) (0.0027) (0.0199) ww β11 0.0350 0.0347 σν3 0.0100 0.0100 σξ3 0.3162 0.3166 (0.0017) (0.0009) (0.0056) ww β12 -0.0150 -0.0149 ρν 0.0000 -0.0509 (0.0023) (0.1088) Note: Standard deviations are in parentheses. Table 2.B.6: Result of Simulation I (J = 3, M = 2, θ12 = θ13 = 0) True ¯ MSL True ¯ MSL True ¯ MSL θ θ θ̂ θ θ θ̂ θ θ θ̂ ww β0 10.0000 9.9999 β11 0.0350 0.0348 σν2 0.0100 0.0095 (0.0348) (0.0015) (0.0024) β1y 0.4000 0.3994 ww β12 -0.0150 -0.0149 σν3 0.0100 0.0100 (0.0139) (0.0019) (0.0005) β2y 0.3000 0.3001 ww β22 0.0250 0.0250 ρν 0.0000 -0.0445 (0.0129) (0.0032) (0.0992) yy yw β11 0.0500 0.0500 β11 0.0100 0.0100 θ12 0.0000 0.0005 (0.0038) (0.0007) (0.1171) yy yw β12 -0.0100 -0.0100 β12 -0.0050 -0.0050 θ13 0.0000 -0.0009 (0.0033) (0.0011) (0.1200) yy yw β22 0.0400 0.0400 β21 0.0150 0.0150 ρ -0.5000 -0.4996 (0.0032) (0.0004) (0.0116) yw β1w 0.2500 0.2496 β22 -0.0050 -0.0050 σT 0.2236 0.2243 (0.0026) (0.0007) (0.0172) β2w 0.4000 0.4003 σv 0.3162 0.3150 σξ2 0.3162 0.3163 (0.0034) (0.0071) (0.0073) σξ3 0.3162 0.3165 (0.0040) Note: Standard deviations are in parentheses. 74 CHAPTER 3 DEMAND ESTIMATION OF DEPOSITS: A CASE OF THE KOREAN FINANCIAL INDUSTRY 3.1 Introduction During the last several decades, tools in structural economic modeling have developed re- markably, especially in the eld of industrial organization. These techniques and tools were 1 recently applied in nance to some extent, presenting many promising directions. For ex- ample, Hortaçsu et al. (2018) estimate a structural model of the uniform price auctions of U.S. Treasury bills and notes in order to analyze market power across the three dierent bidder groups: primary dealers, direct bidders, and indirect bidders. Bonaldi et al. (2015) propose a framework for estimating spillover eects between individual banks' short-term funding costs and measure systemic risk using data from the main renancing operations of the European Central Bank. The other intersecting eld of nance and industrial organization is estimating a demand system for nancial assets, which are viewed as dierentiated products. There are two main directions with respect to this eld: the rst one is based on a product-space demand model like the one proposed by Deaton and Muellbauer (1980) that approximates the demand function by a exible functional form; and the second direction is based on a characteristics space demand model such as Berry et al. (1995, hereafter BLP), where consumer choices are based on products' characteristics rather than the products themselves. The aim of this paper is to estimate a structural demand model for the nancial instru- ments of Korea in order to measure the eect of deregulation in the payment and settlement systems. From 2009, securities companies were given access to participate in retail payment systems, which were previously restricted to banks only. Consequently, cash management ac- 1 Please refer to Kastl (2017) for more details. 75 counts (hereafter CMAs) provided by securities companies, which were similar to traditional deposits of banks but had limitations in transferring funds, became the close substitutes for deposits in terms of services. CMAs, which were introduced in Korea in the 1980s, have similarities to the checking accounts of banks that consumers can deposit and withdraw funds from without limitations. In addition, as securities companies generally invest funds from CMAs in government or 2 public corporations bonds with repurchase agreements , they oer interest rates of CMAs around the policy interest rates, whereas the checking accounts usually provide almost zero interest. That is, CMAs share the features of the checking accounts and the time deposits of banks. However, securities companies are regarded as less safe than banks due to the dierences in the business model and the size of institutions, as well as regulatory gaps between banks and securities companies. Also, most CMAs are not protected by deposit 3 insurance. Furthermore, as the retail payment systems were only accessible by banks, CMAs were not used as a means of exchange. Ever since the Capital Market and Financial Investment Business Act, enacted in August 2007, was enforced in February 2009, securities companies were allowed to participate in the retail payment systems operated by the Korea Financial Telecommunications and Clearings 4 Institute (KFTC). Therefore, from the depositor's perspective, traditional deposits and CMAs became indistinguishable in terms of services they provide. For instance, consumers who have CMAs are able to transfer funds to bank accounts via internet or mobile banking services and vice versa. Also, CMA holders can pay o their credit card balances by deducting 5 from their CMAs. Reecting these changes, CMAs were included in M2 from July 2009. 2 As of Q4 2016, CMAs with RP agreement count for 59.0% of total CMAs, while those investing in MMF and duciarily managed by Korea Securities Finance Corporation count for 6.1% and 30.3%, respectively. 3 CMAs of a securities company that also has the merchant bank license are protected by deposit insur- ance. However, its share of CMAs is only 4.6% as of Q4 2016 since only two securities companies hold the merchant bank license. 4 Securities companies began to join the retail payment systems from July 2009. 5 Although CMAs are transferable like the checking accounts included in M1, depositors in CMAs have to sacrice interests if they use balance in CMAs for transaction. Therefore, CMAs are classied into M2. Please refer to International Monetary Fund (2016) for more details on the denitions of money aggregates. 76 The enactment of the Capital Market and Financial Investment Business Act sparked a erce debate on securities companies' participation in the retail payment systems. People who supported the deregulation claimed that consumer welfare would be improved by pro- moting competition among nancial institutions, and it would be necessary to promote the nancial investment businesses that were less developed than the banking industry. On the other hand, people who were against the measure contended that it would be harmful to the nancial system as it would cause an increase in payment and settlement risks. Also, 6 receiving deposits is considered the banks' own business and only a few countries allowed securities companies to participate in the retail payment systems. As a result, the Capital Market and Financial Investment Business Act permitted securi- ties companies to participate in the retail payment systems. However, although CMAs have the advantages of interest rates and services compared to bank deposits, its total amount has stabilized after a sharp increase between mid-2006 and mid-2008. In particular, its balance had remained around forty trillion KRW for ve years since the global nancial crisis. This suggests that depositors' choice may depend on the nancial stability situation that would aect their risk attitudes, referred to as the market discipline in banking. Based on this phenomenon, this paper evaluates whether consumer welfare has signicantly increased with the enforcement of the act when considering consumer's risk attitudes. In order to measure the eect of deregulation, I develop a structural demand model following the characteristic space approach. As in Petrin (2002), the researcher is able to evaluate welfare gains for consumers from the introduction of new products by constructing a structural model. Furthermore, as Nevo (2000) notes, the econometrician can reduce the number of parameters that need to be estimated. To estimate the model, I apply the random coecient discrete choice approach. This approach can estimate the model using only market-level price and quantity data, deal with the price endogeneity, and allow for a 6 For example, the U.S. Bank Holding Companies Act denes banks as an institution which both (i) accepts demand deposits or deposits that the depositor may withdraw by check or similar means for payment to third parties or others; and (ii) is engaged in the business of making commercial loans. 77 Figure 3.1.1: Interest Rates and Total Amount of CMAs (a) Interest Rates (b) Total Amount Source: Bank of Korea, Korea Financial Investment Association more realistic substitution pattern reecting the heterogeneity in consumer tastes. The characteristics space approach model in the nance literature relates to asset pricing and portfolio choice. Markowitz (1952), the classical reference in nance, views a portfolio as bundles consisting of mean-variance characteristics. In addition to the mean and the vari- ance of returns, other relevant characteristics of nancial instruments may include maturity, probability of default, asset covariance with the market return, etc. However, as Kastl (2017) points out, although those might be the relevant characteristics that capture important parts of variation in demand for portfolio, it might be hard to succinctly capture other important ones. Other than analyzing portfolio choice that considers whole nancial markets, another way of dening the relevant characteristics is by restricting the scope of the nancial instru- ments, such as deposits. This paper focuses on deposits instruments, which include checking, 7 savings, and time deposit accounts generally provided by commercial banks, and CMAs. The reasons are as follows: (i) the banking sector holds more than 50% of the nancial 8 assets among Korean nancial institutions ; (ii) deposits are the major source of Korean 7 One can consider to separately construct models by products. However, as deposits cannot be disag- gregated at the bank level as well as CMAs hold characteristics of both checking and time deposits, I focus on whole deposit services. 8 As of Q4 2016, the banking sector holds 50.8% of the nancial assets among Korean nancial institutions, while insurance sector and securities sector hold 15.9% and 5.8%, respectively. 78 9 banks' funding , and (iii) CMAs, the interest of this paper, became the close substitute for traditional deposits by the deregulation in payment and settlement systems. Recently, some papers have applied a discrete choice model to estimate the demand for deposits. For example, Dick (2008) estimates a structural demand model for commercial bank deposit services in order to measure the eects on consumers, given changes in bank services owing to the Riegle-Neal Interstate Banking and Branching Eciency Act of 1994 that allowed for nationwide branching. Following the discrete choice literature, it assumes that consumer decisions are based on prices and bank characteristics, such as deposit rates, account fees, the age, size, and geographic diversication. Based on the demand estimation for deposits, some papers extended a structural model of the banking sector to analyze the nancial fragility. For example, Egan et al. (2017) develop a structural empirical model of the U.S. banking sector that considers both demand and supply sides. After estimating the demand and supply for deposit, the researchers evaluate several proposed bank regulations. The results, for instance, suggest a capital requirement below eighteen percent could lead to signicant instability in the U.S. banking system. However, those papers do not explicitly take market discipline in banking into consid- eration. Market discipline in banking, in its broad terms, is dened as the mechanism via which market participants monitor and discipline excessive risk-taking behavior by banks (Stephanou, 2010). It is often described as a situation where depositors face costs that are positively related to bank risk and react on the basis of these costs (Berger, 1991). For instance, given that the bank's fragility increases, depositors respond by withdrawing their funds or by demanding higher interest rates on their deposits. Since it is known that mar- ket discipline would lower the probability of individual bank's failures and the incidence of banking crises by reducing the problems of moral hazard and asymmetric information in banking, policymakers have increasingly recognized its role and have incorporated it in their regulatory frameworks. One example of its codication is Pillar 3 in the Basel III, which is 9 As of Q4 2016, deposits consists of 83.6% of banks' funding. 79 the global supervisory framework for internationally active banks. 10 Much work has been done on the existence of market discipline. Previous studies provide evidence of market discipline in both developed countries and developing countries. Most of these studies examine the existence of market discipline by analyzing either how yields on uninsured deposits or the level or growth of uninsured deposits respond to measures of bank risk. However, a number of papers have found that the typical test for existence of market discipline might fail in some developing economies in non-crisis periods, as traditional indicators of bank soundness tend to become less signicant and explain a smaller fraction of the total variance of deposits and interest rates during nancial turmoil than during stable periods. The results imply that depositors behave dierently by the nancial stability situation. The remainder of the chapter is organized as follows. Section 3.2 outlines the model specications and estimation strategies. Section 3.3 describes the data and instruments. Section 3.4 reports the estimation results. Section 3.5 concludes the chapter. 3.2 Empirical Framework 3.2.1 Assumptions 11 I assume that, following Dick (2008), consumers cluster their deposits within one primary bank for acquiring banking services together. Based on this assumption, one can apply the discrete choice model. It might be possible for consumers to demand multiple banking services. However, if banks were to provide benets to depositors who use the bank as 12 the primary one, which is common in Korea , consumers would then have incentives to 10 Please refer to Flannery (1998), Arena (2003), and Levy-Yeyati et al. (2010) for more details. 11 Due to the limitation of data that it does not divide depositors into households and corporates by nancial institutions, I assume that two groups of depositors choose a depository institution in a similar manner. Dick (2008) also assumes that their behavior is similar based on the consumer and business survey. 12 For instance, banks oer higher deposit interest rates and lower fees on transactions to depositors depending on their class, which is decided by the amount of deposit, the records of direct deposit of salary, the number of accounts, etc. 80 consolidate their deposits in a single nancial institution. In addition, according to the 13 Survey of Household Finances and Living Conditions , the median amount of deposits per household is thirty three million KRW as of the end of March 2017, which is lower than the amount of deposits protected by deposit insurance (fty million KRW). These suggest that it is reasonable to assume that consumers choose a single bank for deposits. Given that CMAs have become the close substitutes since the deregulation, securities companies providing them are assumed to be treated as banks in the deposits market, albeit it seems to be a strong assumption. I dene market share based on the amount of deposits, and outside goods as deposits in 14 nancial institutions other than banks and securities companies; these include merchant banking corporations, mutual savings banks, credit cooperatives, and postal savings. This implies, along with the rst assumption, that depositors can have a number of accounts as long as they cluster deposits into one bank. The denition of market share using the amount of deposits, not the number of accounts, makes up for the shortcomings of the rst assumption, which enables to apply a discrete choice model within a multinomial choice setting. For instance, even though consumers hold accounts in multiple banks, the problems that stem from the rst assumption could be mitigated as long as the amount of deposits in banks other than the primary one is negligible. In addition, given that transferring funds is easier to do than opening and closing accounts, it will reduce the xed cost to change one's primary bank if consumers have accounts in multiple banks. The denition of outside goods has limitations as it might not capture the true market share since some people may choose to invest funds in nancial instruments other than deposits. However, the results of the Survey of Household Finances and Living Conditions, which shows that households' preference for nancial instruments have remained stable, suggest that this study's denition of outside goods would therefore be reasonable. 13 The survey is annually conducted of twenty thousand households by the Statistics Korea, the Financial Supervisory Service of Korea, and the Bank of Korea since 2012. 14 I exclude KDB and KEXIM from the category of banks due to their heterogeneous business model. 81 Table 3.2.1: Households' Preferences for Financial Instruments Deposits Pension Stock Etc. Total 2012 89.9 1.7 5.9 2.5 100.0 2013 90.7 1.8 4.7 2.8 100.0 2014 91.6 2.2 3.4 2.8 100.0 2015 90.6 2.3 4.7 2.4 100.0 2016 91.6 1.9 4.0 2.5 100.0 2017 91.8 1.8 4.1 2.3 100.0 Source: Statistics Korea, Financial Supervisory Service of Korea, and Bank of Korea, Survey of Household Finances and Living Conditions 3.2.2 Models In the characteristics space demand model, the price of a product can be correlated with an omitted product attribute, which is relevant but not observed by the econometrician. If an omitted product attribute is positively correlated with the price, estimates of the price sensitivity term will be biased toward zero and those of the price elasticities will be biased 15 as well. To deal with the potential price endogeneity problem, one can use instrumental variables and/or apply a random coecient discrete choice model. Thus, I construct the following models to estimate the demand for deposits: (i) the simple conditional logit model that does not include an omitted product attribute (hereafter Conditional Logit); (ii) the Berry (1994) type logit model that includes an omitted product attribute (hereafter IV Logit); (iii) the simple random coecients logit model that does not include an omitted product attribute (hereafter RC Logit); and (iv) the BLP (1995) type random coecient logit model (hereafter BLP (1995) RC Logit). 3.2.2.1 Conditional Logit and IV Logit Models Similar to most discrete choice models following the Random Utility Maximization (RUM) hypothesis, I assume that individual agents i = 1, . . . , I (= ∞) at t = 1, . . . , T markets make choices between j = 1, ..., J alternatives in order to maximize their indirect utility, 15 Kim and Petrin (2015) provide a literature review about this problem. 82 uijt , specied as uijt = x′jt β + αpjt + ξjt + ϵijt = δjt + ϵijt , where xjt = (xjt,1 , . . . , xjt,K )′ is a K ×1 vector of observed characteristics for deposit product j at the market t, pjt is the spread or interest rates paid by banks on j at t, ξjt is an unobserved characteristic for j at t, and ϵijt is the error term. As Conditional Logit model does not take account for unobserved heterogeneity, ξjt = 0 for all j and t. δjt = x′jt β + αpjt + ξjt is referred to as the mean utility, which is common to all agents. The K + 1 dimensional vector θ = (β, α) represents the taste parameters. Now, assume that ϵijt are identically and independently distributed according to the Type I extreme-value distribution. Then, by integrating over ϵijt , the predicted market share for j at t is derived such that ′ exp(xjt β + αpjt + ξjt ) sjt (x, β, α, ξ) = PJ ′ . (3.2.1) r=1 exp(xrt β + αprt + ξrt ) Berry (1994) assumes that at the true parameter values, β0 and α0 , the following equality must hold sjt (x, β0 , α0 , ξ) = Sjt , where Sjt is the true market share from the aggregated data. In other words, conditioning on the true values of δ0 , the model should exactly t the data. Berry (1994) uses the following transformation of equation (3.2.1) such that log(sjt (x, β, α, ξ)) = et + x′jt β + αpjt + ξjt , et = −log( Jr=1 exp(x′rt β +αprt +ξrt )). P where By normalizing the mean utility of the outside good, denoted as j = 0, to zero that implies log(s0t (x, β, α, ξ)) = et , 83 equation (3.2.2) is obtained such that log(Sjt ) − log(S0t ) = δjt = x′jt β + αpjt + ξjt , (3.2.2) where S0t is the share of the outside good at t. Given equation (3.2.2), one can estimate the Conditional Logit model with ordinary least squares by regressing log(Sjt ) − log(S0t ) on (x′jt , pjt ), as well as IV Logit model with instrumental variables estimation given the assumption E[ξjt |Zjt ] = 0. 3.2.2.2 RC Logit and BLP (1995) RC Logit Models For RC Logit and BLP (1995) RC Logit models, I specify the indirect utility similar to Nevo (2000) that allows the price coecient to be random without taking the natural log. Therefore, the indirect utility of an agent i from consuming j at the market t is specied as uijt = xjt βi + αi pjt + ξjt + ϵijt , where βi,k = βk +σk ηi,k , αi = α+σp ηi,p , ηi,k , ηi,p ∼ N (0, 1), ξjt is an unobserved characteristic for j at t, and ϵijt is the error term. Now, I decompose indirect utility by two parts: the mean utility, δjt , and the het- eroskedastic error terms, νijt , that captures the eect of random tastes parameters such that uijt = δjt + νijt , P where δjt = xjt β + αpjt + ξjt represents a mean level of utility and νijt = [ k xjt,k σk ηi,k ] + σp ηi,p pjt + ϵijt represents a heteroskedastic error terms that captures the eect of random tastes parameters. In order to estimate the model, I dene the set of values of error terms, Ajt , that make j maximizing utility at t given the J dimensional vector δt = (δ1t , . . . , δJt ), such that Ajt (δt ) = {νit = (νijt ) | δjt + νijt > δj ′ t + νij ′ t , ∀ j ′ ̸= j}. 84 Then, the market share for j at t is written as Z sjt (δt (x, p, ξ), x, β, α, σ) = f (ν)dν. Ajt (δt ) In order to estimate the models, I take the following steps. First, I compute the market shares given δt and σ such that Z P exp(δjt + k xjt,k σk ηi,k + σp ηi,p pjt ) sjt (δt , σ) = PJ P df (ηi ). 1+ r=1 exp (δrt + k x rt,k σ k η i,k + σ p ηi,p p rt ) Second, given σ, I nd δjt by contraction mapping. Third, given δjt , β , and α, obtain ξjt . Last, choose β, α and σ to minimize the sample criterion function. For example, I use the moment condition of E[ξjt (β0 , α0 , σ0 )|Zjt ] = 0 to estimate BLP (1995) RC Logit by GMM. 3.3 Data and Instruments 3.3.1 Data The data mainly come from two sources: nancial institution-level data from the Finan- cial Statistics Information System (FISIS) of the Financial Supervisory Service of Korea (http://sis.fss.or.kr), and country-level aggregate data from the Economic Statistics System (ECOS) of the Bank of Korea (http://ecos.bok.or.kr). The data on each nancial institu- tion's deposits and its attributes are obtained from the balance sheet, the income statement, and other reporting forms uploaded on FISIS. The data on the total amount of deposits from the Flow of Funds and the policy interest rates are taken from ECOS. The amount of CMAs is obtained from the Korea Financial Investment Association Portal (FreeSIS, http://freesis.koa.or.kr). The sample covers the period from Q1 2003 to Q2 2015 consid- ering the completion of the restructuring Korean nancial industry after the Asian nancial crisis (Q4 2002), the enforcement of Capital Market and Financial Investment Business Act (Q2 2009), and the merger of Hana and KEB banks (Q3 2015). 85 16 An observation is dened as a nancial institution - quarter combination in the estima- tion exercises. I choose the attributes of nancial institutions from available data, which are important and easily observable by depositors. Table 3.B.4 shows summary statistics of data. I use spread, which is the dierence between interest rates paid on deposits and the policy interest rates, as the price variable. This is because deposit rates are decided in line with the policy interest rates and the interest rates regime shifts before and after the global nancial crisis. The deposit rates are driven by dividing interest expense on deposits by the amount of deposits from each institution's quarterly income statement and annualized. In addition to the price variable, four categories of observed characteristics are chosen: (i) size, (ii) quality of service, (iii) quantity of service, and (iv) nancial soundness. Similar to Dick (2008), I classify nancial institutions into ve groups, considering their asset sizes and 17 other characteristics , and use them to control for size rather than using the asset size itself. The reason is that the asset size itself should increase as the nancial institution receives 18 more deposits by the law of accounting. In addition, it would capture features associated with the size of nancial institutions, including larger infrastructures, product diversity, and know-how. The quality and quantity of service are proxied by the number of employees per 19 20 branches and the number of branch , respectively. I include the nancial soundness indicator and a dummy variable for the period of - nancial turmoil in order to test the existence of market discipline in the deposit market. Egan et al. (2017) use the implied probability of default of banks from credit default swap (CDS) spreads when estimating the demand for deposits. However, it is not easily available 16 As the amount of CMAs of each securities rm is not available in public, I assume them as a single entity. 17 For more details, please refer to Table 3.B.5. 18 For the sample period, the correlation between dependent variable and asset size is 0.91. 19 Dick (2008) argues that it can capture consumers' waiting time, the types of services specic to bank, and the value of human interaction to consumers who are not able to use the online service. 20 One can consider the number of ATMs as a proxy for the quantity of service. However, the data on the banks' number of ATMs does not cover the whole sample period as well as these on securities companies are not provided. Therefore, I do not include the number of branch although it seems to be relevant. 86 to depositors, and its value might highly depend on the model and assumptions. Therefore, I use the risk-based capital ratios, which are representative, well-known, and publicly dis- closed indicators: the BIS ratio for banks and the net operating capital ratio for securities 21 companies. I assume the period from Q3 2008 to Q2 2013 as a time of nancial instability reecting major nancial events and nancial stability indices. 3.3.2 Instruments I use three categories of instrumental variables: (i) nancial institutions' characteristics themselves; (ii) mark-up shifters; and (iii) cost shifters. The set of mark-up shifters includes BLP instruments, which are the sum of characteristics of other products in the market, following the convention of the literature on discrete choice models. This is based on the intuition from models of oligopoly that suggest the more isolated the rm is in the product 22 space, the more likely it is to have a higher price relative to the cost . The set of cost shifters includes variables related to marginal costs, funding costs and labor costs. I use the policy interest rates as a proxy for funding costs, as the interest rates of funding sources other than deposits, such as bank debenture and call money, are also decided based on it. Labor costs come from the average wage data of the nancial business from Statistics Korea. These two variables are chosen, although the income statement provides data for each nancial institution, because the nancial institution's technology and qual- ity are already controlled through other covariates. For example, if a nancial institution hires more skilled workers whose wages tend to be higher than those of low-skilled workers, the actual salary data may contain the hidden quality components, therefore leading it to violating the independent assumption. 21 I exclude the credit card debacle in 2003, since the problem stemming from credit card companies might not aect depositors' risk attitude as well as the debacle was recovered in the short time. 22 Please refer to BLP (1995) for more details. 87 3.4 Results 3.4.1 Model Estimation Table 3.B.7 presents the estimation results, where column (1) corresponds to the Conditional Logit model, columns (2) and (3) correspond to the IV Logit model, column (4) corresponds to the RC Logit model, and columns (5) and (6) correspond to the BLP (1995) RC Logit model. In columns (2) and (5), nancial institution's characteristics themselves and mark- up shifters are used as instrumental variables, whereas cost shifters are included as well in columns (3) and (6). Coecients of RC Logit and BLP (1995) RC Logit are the mean values of random coecients (β, α). In order to test whether the existence of market discipline depends on the nancial stability situation, the nancial turmoil dummy variable interacts with both the spread and the risk-based capital ratios. In addition, since the risk-based capital ratios between banks and securities companies are dierent, an additional dummy variable that represents securities companies interacts with them. The results from the Conditional Logit model show that the coecients on spread in both the stable period and the nancially distressed period are signicantly negative, im- plying that an unobserved attribute is correlated with spread; thus, it is biased toward zero. Although the random coecient model is known to deal with the price endogeneity problem, the result in column (4) shows that the coecient on spread is statistically insignicant in the stable period while its sign is reversed to positive. However, the results from the IV Logit and the BLP (1995) RC Logit models for which the price variable is instrumented show that the spread coecients in the stable period have the expected sign and are statistically signif- icant. Furthermore, the magnitude of coecient substantially increases in the BLP (1995) RC Logit model compared to that in the IV Logit model. This is in line with the nding from related studies (e.g., BLP (1995), Petrin (2002)). Table 3.B.6 shows the distribution of own-price elasticities for the tranquil times obtained from the IV Logit model and the BLP 88 (1995) RC Logit model. The coecients on spread as well as the risk-based capital ratio in nancial turmoil sug- gest that the market discipline in Korean banking sector has appeared dierently depending on the nancial stability situation. In both the IV Logit model and the BLP (1995) RC Logit model, the coecients on spread are not signicantly dierent from zero during the period of 23 nancial instability. Instead, the coecients on the banks' risk-based capital ratio become signicantly positive, whereas those in the stable period are signicantly negative. That is, regardless of the deposit rates, consumers prefer to deposit in a safer depository institution when the nancial system is unstable. This phenomenon is similar to the ight to quality in the bond and equity markets occurred during the nancial crisis. It is counter-intuitive that the coecients on the risk-based capital ratio are negative in tranquil times. However, for instance, the BIS ratios of Korean banks have been maintained over the minimum requirement during the sample period due to the experience of the Asian nancial crisis. Figure 3.4.1 represents the unweighted BIS ratios of Korean banks and the minimum requirement. Therefore, it would be possible that depositors might regard them as safe regardless of the level of the BIS ratio. In addition, if consumers with low credit scores can use other services provided by a bank, such as loans, by depositing, it would lead to a lower risk-based capital ratio of the bank. The signs, magnitudes, and signicance of other coecients are in accord with expec- tations. Depositors respond favorably to the size, the branch stang, and the number of branches of depository institutions. The result that the coecients on Group 2 nancial institutions are signicantly positive in the BLP (1995) RC Logit model reects the charac- teristics of banks in the group: one is specialized in the transaction of foreign exchange, and the other is established in order to support nancing of small to medium enterprises. The reasons for the negative coecients on Group 5 nancial institutions, securities companies, 23 The coecients on the securities companies' risk-based capital ratio are statistically insignicant in nancial turmoil. However, considering that those in the stable period are negative and depositors in CMAs would have dierent risk attitude from depositors in banks, one can interpret this that depositors, even who are less risk-averse, become more risk-averse in the times of nancial instability. 89 Figure 3.4.1: BIS ratios of Korean Banks Source: Financial Supervisory Service of Korea seems to be (i) dierences in institutional framework from banks, including deposit insurance and regulation; and (ii) stigma eects from the collapse of Dongyang Securities whose CMAs 24 market share was one of the highest before the bankruptcy. 3.4.2 Consumer Welfare In order to measure the eect of deregulation on consumer welfare, I calculate the equivalent variation (EV ) following Small and Rosen (1981) in the context of the discrete choice model. According to Dick (2008), the equivalent variation (EV ) can be calculated as EV = St (p, x; θ) − St−1 (p, x; θ), (3.4.1) where S(p, x; θ) = ln[Σj exp(δj (pj , xj ; θ))]/α, and δj = x′j β + αpj + ξj , θ = (β, α). 25 However, the estimation results show that depositors do not respond to spread in the times of nancial turmoil. This implies that even though CMAs oer higher interests than 26 banks that might induce banks to increase the deposit rates they oer, the deregulation 24 Although CMAs balance for each securities company is not disclosed, it was known that CMAs balance of Dongyang Securities was around 10 trillion KRW in the peak. 25 In this formula, α, the coecient on spread, represents the marginal utility of income. 26 During the sample period, the average spread of securities rms is 6.4 bps, whereas that of banks is -41.9 bps. 90 may not aect the consumer welfare at all in terms of a monetary unit. Furthermore, one cannot exclude the possibility that it might have a negative eect on consumer welfare due to the weakness of CMAs or securities companies illustrated in Section 3.1: (i) most CMAs are not included in the scope of the nancial instruments protected by the deposit insurance and (ii) there exists a regulatory gap between banks and securities companies. Therefore, I compare changes in welfare focusing on the stable period following equation (3.4.1). Depositors experience a gain in welfare due to deregulation between tranquil times, with a mean of KRW 0.0005-0.005 per consumer per year. This implies, for example, with a welfare gain of KRW 0.002, a depositor carrying a median balance (33 million KRW as of the end of March 2017) can gain 66 thousand KRW per year. However, it should be noted that the welfare gain has been diluted due to the prolonged nancial stress since the global nancial crisis. 3.5 Conclusion This paper sought to apply structural econometric modeling in the eld of industrial or- ganization to nance. The results suggest that unlike other products (e.g., automobiles, Petrin (2002)), a new nancial instrument does not necessarily improve consumer welfare even if it seems competitive in terms of price; this nding may be due to the existence of market discipline within nancial markets. This implies that in order to achieve the goal of deregulation in the payment and settlement systems, it is necessary to devise an institutional framework that can reduce the dierence in risk between products and nancial institutions, which would foster a level playing eld for nancial institutions. The model of the paper relies on simplifying assumptions. For instance, given that the services provided by securities companies are dierent from banks, the assumption of treating securities companies that provide CMAs as banks might not reect the reality. Also, some consumers might split a signicant amount of deposits in multiple banks. To manage the 91 problem, one can consider applying multiple-discrete choice model (e.g., Hendel (1999)), which may need micro-level data. It is important to note that the approach in this paper uses only market-level data. Having taken basic but important steps in estimating a demand system, this model has the potential to lead to future research with improvements. For example, the model can be used to measure the eect of changes in prudential regulation. Also, given that two internet-only banks were newly established in Korea in 2017, the demand model for deposits taking account of both price and service competition can be extended to measure the eect of introducing internet-only banks. 92 APPENDICES 93 APPENDIX A An Overview on the Korean Financial System 3.A.1 Financial Industry The Korean nancial system has been developed as bank-based, in that banks play a leading role in mobilizing savings, allocating capital, overseeing the investment decisions of corporate 27 managers, and providing risk management vehicles. Table 3.A.1 shows the total assets of the major nancial institutions in Korea and their shares. Although banks' asset shares in the nancial system have decreased after the Asian nancial crisis, they still account for the largest portion with more than 50%. Table 3.A.1: Total Assets of Major Financial Institutions in Korea (Unit: Trillion KRW, %) 1990 1995 2000 2005 2010 2015 249.7 595.8 982.2 1,213.5 1,884.1 2,440.7 Banks (63.3) (62.9) (63.4) (57.8) (54.8) (57.3) Merchant Banking 23.7 45.9 21.3 13.2 24.2 11.1 Corporations1) (6.0) (4.8) (1.4) (0.6) (0.7) (0.3) 11.5 32.6 24.2 44.9 91.3 43.9 Mutual Savings Banks (2.9) (3.4) (1.6) (2.1) (2.7) (1.0) 24.6 75.9 145.1 220.2 360.9 533.5 Credit Cooperatives (6.2) (8.0) (9.4) (10.5) (10.5) (12.5) 3.4 7.0 24.5 37.8 55.4 65.6 Postal Savings (0.9) (0.7) (1.6) (1.8) (1.6) (1.5) 34.6 86.6 163.6 308.6 507.5 816.0 Insurance Companies (8.8) (9.1) (10.6) (14.7) (14.8) (19.2) 16.6 27.8 42.0 62.7 189.4 344.5 Securities Companies (4.2) (2.9) (2.7) (3.0) (5.5) (8.1) Collective Investment 30.6 76.0 146.7 198.4 325.3 4.8 Business Entities (7.8) (8.0) (9.5) (9.4) (9.5) (0.1) 394.8 947.6 1,549.5 2,099.2 3,438.0 4,260.1 Total (100.0) (100.0) (100.0) (100.0) (100.0) (100.0) Note: 1) Including consolidated nancial accounts of banks and securities companies. Source: Bank of Korea, Financial Supervisory Service of Korea Since the 1980s, the Korean government had eased regulations on the nancial market entry in order to foster competition among nancial institutions. As a result, the number 27 Please refer to Demirgüç-Kunt and Levine (1999) for more details on bank-based and market-based nancial systems. 94 of banks increased to thirty-three before the Asian nancial crisis in 1997. However, as the soundness of banks deteriorated during the crisis, insolvent nancial institutions were resolved through liquidation or mergers and acquisitions based upon judgements as to their survivability. Therefore, as of Q4 2002 when restructuring due to the crisis was nalized, 28 the number of banks decreased to nineteen. As of Q4 2016 , there were seventeen banks, 29 including six nationwide banks, six local banks, and ve specialized banks. Table 3.A.2 shows the list of banks in Korea. Table 3.A.2: List of Banks in Korea (as of Q4 2016) Nationalwide Banks Local Banks Specialized Banks Kookmin Kyongnam Nonghyup Shinhan Kwangju Suhyup Woori Daegu IBK KEB Hana Busan KDB SC Korea Jeonbuk KEXIM Citibank Korea Jeju Source: Financial Supervisory Service of Korea Deposits are major funding sources for Korean Banks. To be specic, as of Q4 2016, deposits consisted of 83.6% of banks' funding. Therefore, banks have the largest portion, 68.4%, in the deposit market, and it is made up of deposits in depository institutions, such as banks, merchant banking corporations, mutual savings banks, credit cooperatives, postal savings, and CMAs. 3.A.2 Payment and Settlement Systems The payment and settlement systems in Korea consist of a large-value payment system, retail payment systems, securities settlement systems, and foreign exchange settlement systems. While a large-value payment system is used for transactions between nancial institutions, 28 Shinhan and Chohung were merged in Q2 2006, and Hana and KEB were merged in Q3 2015. 29 Specialized banks are established with specic purposes of bolstering nancing in areas encountering funding diculties due to shortages of nance, protability and expertise. However, except Korea Develop- ment Bank (KDB) and Export-Import Bank of Korea (KEXIM), their business model, such as the funding structure, is similar to commercial banks. 95 retail payment systems are used for those among individuals or corporations. By the en- forcement of the Capital Market and Financial Investment Business Act, as of Q4 2016, 30 twenty-ve securities companies are participating in six retail payment systems operated 31 by KFTC . Thus, CMA holders became able to use them as a means of exchange. However, retail payment systems are processed by net settlements that net obligations arising from transactions in the retail payment systems are transferred between the current accounts of the nancial institutions involved at a designated time. Therefore, unlike the real-time gross settlement system, nancial institutions are exposed to settlement risks such as credit risk when the counterpart fails to transfer fund. 3.A.3 Financial Stability Situation In order to test the hypothesis that the existence of market discipline depends on the nancial stability condition, it is essential to identify a period of nancial distress. However, as Aspachs-Bracons et al. (2012) point out, it is hard to measure nancial fragility, whereas ination can be measured by a relatively simple and intuitive variable, the consumer price index. After the global nancial crisis, there has been a growth in literature concerning the 32 eld of devising the nancial stability index. Figure 3.A.1 shows the nancial stability indices of Korea, where (a) represents a com- posite nancial stability index published in the Financial Stability Report of the Bank of Korea, (b) represents CoVaR based on Adrian and Brunnermeier (2016), and (c) represents Marginal Expected Shortfall (MES) based on Tarashev et al. (2010). It shows that these indices all have similar trends reecting the major events in the Korean nancial system. Table 3.A.3 shows major nancial events in Korea since 2000. 30 Sixteen major securities companies joined the retail payment systems in Q3 2009. 31 These include Electronic Banking System, Cash Management Service Network, Interbank Remittance System, Giro System, CD Network, and Payment Gateway System. As of Q4 2016, those systems counts for 89.2% of total transaction volume. 32 Please refer to Silva et al. (2017) for more details. 96 Figure 3.A.1: Financial Stability Indices of Korea (a) Composite Index (b) CoVaR (c) MES Source: Bank of Korea Financial Stability Report, Lee et al. (2013) Table 3.A.3: Major Financial Events in Korea Period Events 2003 Credit Card Debacle 2008- Global Financial Crisis 2010- European Debt Crisis 2011 Bankruptcy of Mutual Savings Banks 2013 Bankruptcy of Dongyang Securities 97 APPENDIX B Tables Table 3.B.4: Summary Statistics Mean Std. Dev. Max Min Market Share 0.0412 0.0394 0.1604 0.0014 Spread -0.0041 0.0073 0.0238 -0.0218 - Stable Period -0.0076 0.0049 0.0042 -0.0218 - Financial Turmoil Period 0.0010 0.0072 0.0238 -0.0186 Deposit Interest Rates 0.0284 0.0070 0.0534 0.0107 Policy Interest Rates 0.0325 0.0096 0.0515 0.0170 Group 2 Financial Institutions 0.1214 0.3267 1 0 Group 3 Financial Institutions 0.1214 0.3267 1 0 Group 4 Financial Institutions 0.4248 0.4946 1 0 Group 5 Financial Institutions 0.0291 0.1683 1 0 Emplyees per Branch 12.6582 3.1767 28.0752 7.2697 Number of Branch 475.5334 411.8725 1,789 31 BIS Ratio 0.1288 0.0183 0.1825 0.0855 - Stable Period 0.1221 0.0172 0.1825 0.0855 - Financial Turmoil Period 0.1389 0.0151 0.1771 0.0940 Net Operating Capital Ratio 0.1654 0.0274 0.2221 0.1217 - Stable Period 0.1379 0.0113 0.1557 0.1217 - Financial Turmoil Period 0.1792 0.0220 0.2221 0.1469 Observations 824 Table 3.B.5: Classication of Financial Institutions Group Description Group 1 Banks with assets over 100 Trillion KRW as of Q2 2015 and belonging to a holding company Group 2 Banks with assets over 100 Trillion KRW as of Q2 2015 (excluding Group 1 banks) Group 3 Banks with assets less than 100 Trillion KRW as of Q2 2015 and foreign owned Group 4 Banks with assets less than 100 Trillion KRW as of Q2 2015 (excluding Group 3 banks) Group 5 Securities Companies Table 3.B.6: Distribution of Own Price Elasticities 10% 25% Median 75% 90% IV Logit Model 0.0512 0.0992 0.1435 0.1950 0.2581 BLP (1995) RC Model 0.2175 0.3067 0.4539 0.6034 0.7599 98 Table 3.B.7: Estimation Results Conditional IV RC BLP (1995) Logit Logit Logit RC Logit (1) (2) (3) (4) (5) (6) Spread ∗∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗∗ - Stable Period -11.9073 18.8223 17.4081 3.5915 49.4622 48.1135 (4.5920) (7.8120) (6.9278) (4.6821) (8.5157) (7.5949) ∗∗∗ ∗∗∗ - Financial Turmoil Period -14.7139 8.9737 4.6535 -19.2045 6.0866 -1.3014 (3.4651) (6.6238) (5.6131) (3.6118) (7.5065) (6.0896) ∗∗ ∗∗∗ ∗∗ Group 2 Financial Institutions -0.1335 0.0946 0.0682 -0.0725 0.2203 0.1748 (0.0640) (0.0773) (0.0709) (0.0876) (0.0848) (0.0758) ∗∗∗ ∗∗ ∗∗ ∗∗∗ ∗ ∗∗ Group 3 Financial Institutions -0.2875 -0.1615 -0.1792 -0.3011 -0.1469 -0.1786 (0.0765) (0.0814) (0.0789) (0.0971) (0.0884) (0.0847) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Group 4 Financial Institutions -0.9126 -0.7756 -0.7925 -1.1262 -0.9514 -0.9753 (0.0996) (0.1088) (0.1053) (0.1124) (0.1159) (0.1110) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Group 5 Financial Institutions -2.9003 -2.6662 -2.6643 -5.4742 -5.1946 -5.2284 (0.3346) (0.3246) (0.3199) (0.9226) (0.3374) (0.3316) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Employees per Branch 0.0706 0.0650 0.0657 0.0652 0.0568 0.0590 (0.0077) (0.0076) (0.0075) (0.0089) (0.0084) (0.0082) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Number of Branch 0.0019 0.0021 0.0020 0.0019 0.0022 0.0021 (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) Risk-based Capital Ratio ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ - Banks -4.2529 -6.5529 -6.5063 -4.2852 -7.5974 -7.6293 × Stable Period (1.2916) (1.2830) (1.2536) (1.3371) (1.4098) (1.3793) ∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗∗ ∗∗∗ - Banks 3.4807 2.7265 3.2595 5.9185 5.0045 5.7892 × Financial Turmoil Period (1.4694) (1.3635) (1.3145) (1.7733) (1.4116) (1.3614) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ - Securities Companies -2.9404 -8.3174 -8.2239 -7.0095 -14.1804 -13.9986 × Stable Period (2.2782) (2.1462) (2.0458) (6.6984) (2.1963) (2.0748) - Securities Companies 1.0803 -1.4192 -0.9217 2.8133 -0.1593 0.7458 × Financial Turmoil Period (1.3535) (1.3286) (1.2767) (4.9401) (1.4447) (1.3664) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Financial Turmoil -0.9872 -1.4363 -1.4870 -1.3268 -1.9962 -2.0884 (0.2588) (0.2442) (0.2313) (0.2971) (0.2594) (0.2473) ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ Constant -3.4206 -3.0279 -3.0279 -2.3475 -1.7301 -1.7213 (0.2171) (0.2082) (0.2041) (0.2147) (0.2290) (0.2238) Note: ∗∗∗ Signicant at 1%, ∗∗ Signicant at 5%, ∗ Signicant at 10%. 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