Chapter 2 An Extension from Network DEA to Copula-Based Network SFA
2.4 Empirical Results
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deviations relative to their sample means, meaning that the sample banks differ substantially in their output quantities and input prices.
Table 1.1 Sample Statistics
Variable Name Mean Standard Dev. Minimum Maximum
Total loans* (
Y
1) 188313 264669 6194 2800000Investments* (
Y
2) 57971 86385 685 919083Noninterest income* (
Y )
3 2220 4442 24 55958Labor (
X
1) 67 83 5 751Physical capital* (
X
2) 5155 7409 41 68274Funds* (
X )
3 248938 334912 12505 3400000Price of labor* (
W
1) 61 18 12 307Price of physical capital (
W
2) 0.3668 0.4866 0.0213 8.8389 Price of funds (W )
3 0.0177 0.0058 0.0008 0.0637*：Measured in thousands of US dollars.
2.4 Empirical Results
We specify both production and cost functions of a bank as the translog form, because it is a flexible functional form providing a second-order approximation, commonly used by numerous practitioners.8 The cost function is required to satisfy the regularity conditions suggested by the fundamental microeconomic theory. The homogeneity and symmetry conditions can be directly imposed on equations (7) and (8), while the monotonicity conditions in prices and outputs and concavity in prices can be verified after the parameters in equations (7) and (8) are estimated.
We extend the copula-based maximum likelihood approach, first proposed by Lai and Huang (2013) under the framework of the SFA, to estimate a 4-equation system,
8 Other functional forms, such as the Cobb-Douglas, constant elasticity of substitution, and generalized Leontief forms, are also potential candidates. Moreover, the production and cost functions may take different forms.
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consisting of equations (1), (7), and (8). The highly non-linear nature of the log-likelihood functions, as mentioned above, challenges empirical researchers.
Kumbhakar and Lovell (2000, p.165) suggest a two-step procedure to consistently estimate the system equations, which is a less efficient but computationally simpler method. We adopt their idea here and conduct simulations to confirm the appropriateness of the method.
In the first step, we view the 4 system equations as the seemingly unrelated regression models. The composed errors in (1) and (4) are assumed to be the standard disturbance terms for the time being, which bias the intercepts of the two equations as the means of the two composed errors are constants differing from zero, by assumption, which are absorbed by the intercepts. The application of the non-linear least squares potentially leads to consistent parameter estimates for the fractional parameters, as well as all of the slope parameters in (1) and (7). Monte Carlo simulations will be conducted and the results are suggestive of consistency for those estimates.
In the second step, these fractional and slope parameters are treated as given, and the share equations of (8) are overlooked since they merely function for identifying the fractional parameters. The constant terms of (1) and (7), along with the parameters that characterize the distributions of v and u and the dependence in the copula function, i.e.,
u1
,v1
,u2
,v2
, and
, are simultaneously estimated with respect to equations (1) and (7) by the copula-based ML approach. Huang et al. (2014) examine the consistency property of the two-step procedure - similar to Kumbhakar and Lovell (2000) in essence - by Monte Carlo simulations under the framework of a semi-parametric regression model. Since the second-step procedure here is analogous to‧
Huang et al. (2014), we shall not carry out simulations once more.
One problem in the second step worth mentioning arises from the treatment of the estimated fractional and slope parameters as given, so that the error terms contain estimation errors that are functions of explanatory variables shown in the first step and hence result in the variances of the error terms being heteroskedastic and the estimated standard errors of the parameter estimates being inconsistent. To correct for the inconsistency, caused by possible misspecification, we suggest using the procedure proposed by White (1982), which requires computing the sandwich-form of the covariance matrix of estimators in order to obtain correct standard errors.9
as the covariance matrix of the estimator ˆ. However, the covariance matrix of the quasi-maximum likelihood estimators has the so-called sandwich form: misspecified models and the derivation of the covariance matrix.
There are two problems to be solved before deriving the partial derivative of lnL/. First, formula Iapp( )A and Iapp( )B contain the sign functions of sign A( ) and sign B( ), which causes the log-likelihood function to be discontinuous with respect to . We re-express lnL for different values of the sign function, i.e., sign( ) 0 and sign( ) 0. Second, it is difficult to calculate the partial derivative of lnL/, since lnL includes the function of 1( ). We adopt the popular software of Mathematica to do the job.
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2.5 Simulation Results
We now briefly describe our design of experiments, followed by evaluating the biasness and mean square error (MSE) of the proposed estimators using Monte Carlo simulations. This helps gain further insight into the performance of our estimators. Here, we specify the production function in the first production stage as the Cobb-Douglas form for simplicity and the cost function in the second stage as the translog form with a single output and three inputs. Following Olson et al. (1980), Fan et al. (1996), and Huang et al. (2014), we consider three sets of variance ratios and variances, i.e. (
, 2)= (1.24, 1.63), (1.66, 1.88), and (0.83, 1.35), where
2
u2
v2 and
u/
v.Table 1.2 presents the simulation results, for the cases of ( ,
1 12)= (1.24, 1.63) and (
2, 22)= (1.66, 1.88), in terms of biases and MSEs for the parameter estimatesof the production function.10 We see that the estimators of
1 and
2 perform well even for the case of the smallest sample size, i.e., N = 300. Their biases and MSEs are quite small and decrease as the sample size increases. The estimator of ln(
1X
1) has similar properties to those of
1 and
2. Although the estimator of ln(
2X
2) has a little larger bias, this bias is relatively small to its true value and diminishes with larger sample sizes, while its MSEs are relatively large, implying that this parameter cannot be accurately estimated. As expected, the estimator of the constant term exhibits quite
10 We also conduct simulations assuming ( , 1 12)= (1.66, 1.88) and ( 2, 22)= (0.83, 1.35); ( , 1 12)= (0.83, 1.35) and ( 2, 22)= (1.24, 1.63). Since the simulation results are quite similar, we choose not to report them to save space.
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large biases, irrespective of the sample sizes. Moreover, its MSEs are also large, which can be attributed to the fact it is confounded with the mean value of the inefficiency term.
Table 1.3 shows the simulation results for the cost parameters. All estimators, except for the intercept, have quite small biases and MSEs. However, the estimator of ln Y has a little larger MSE. The above Monte Carlo simulations provide evidence 1
supporting our proposed estimation procedure in the first estimation step, which relies on the NISUR. Note that the estimators of the constant terms perform poorly and therefore should be revised in the second estimation step. Further recall that we treat the estimates of the fractional and slope parameters as given in the second step.
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Table 1.2 Biases and MSEs of the parameter estimates of the production function
N
300 500 1000 Table 1.3 Biases and MSEs of the parameter estimates of the cost functionN
300 500 1000Having presented that the parameter estimates have the desirable property of consistency, we proceed to perform an empirical study on the U.S. commercial banks.
Tables 1.4 and 1.5 report estimation results from the two-step procedures. Most of the parameter estimates of the production and cost functions achieve the 1% level of significance. Note that the standard errors of
1,
2,
1,
2,
, and the two constant terms are corrected by the sandwich form, as shown in footnote 7. Using these estimates, we verify the regularity conditions required by the microeconomic theory‧ 國
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and find that most of the observations have the correct signs, indicating that those parameter estimates can theoretically characterize the underlying production technologies used in the two production stages.
The fractional parameters of
1 and
2 are estimated to be equal to 0.26 and 0.68, respectively. This tells us that the sample banks allocate respectively 26% and 68% of their entire workforce and capital stock to luring deposits in the first production process. The results appear to be acceptable, since the first production stage mainly involves a fund collection process, where banks are committed to offer safe and convenient banking services to satisfy all customers’ needs. Hence, a bank must accumulate a large amount of tangible assets like branches, ATMs, data processing and storage facilities, security technology equipment, etc. in this stage. The second production stage corresponds to revenue generation, where various funds collected in the first stage are transformed into miscellaneous loans and investments in corporate and government securities, and used to engage in some off-balance sheet activities. In this stage, banks earn their traditional interest revenues and capital gains from loans and investments, as well as non-traditional income, fees, and commissions, from providing financial services to, e.g., credit card holders, wealth management clients, letters of credit issuers in international trade, and new financial derivative products.These services appear to require more labor input. Here, the estimated values of
1 and
2 lead us to infer that a representative bank allocates 74% and 32% of its entire workers and capital, respectively, to fulfill the second production process.‧
Table 1.4 Parameter estimates of the production function
Variable Parameter Estimates Standard Errors
1 0.2618*** 0.0067Note: *** and * denote significance at the 1% and 10% levels, respectively.
Table 1.5 Parameter estimates of the cost function
Variable Parameter Estimates Standard Errors
constant 6.4443*** 0.0101
‧
Note: *** denotes significance at the 1% level.
We use the foregoing parameter estimates to evaluate the individual TE scores in the two production stages, denoted by TE1 and TE2, respectively. The average value of TE1 from the production frontier is equal to 0.6717 with a standard deviation of 0.1127, reflecting that an average bank could be technically efficient if it can produce around 33% more intermediate output. The average TE2 score implied by the cost frontier is equal to 0.7758 with a standard deviation of 0.0919. This implies that an average bank is fully cost efficient, should it cut roughly 28.9% (=1/0.7758 - 1) of its current input mix. Both TE1 and TE2 are accurately estimated due to their small standard deviations relative to the individual means. Our model allows for identifying the fractional parameters and estimating the technical efficiency scores in the two production processes, enabling bank managers to adopt valid strategies to improve operational performance in both stages. Moreover, the value of the correlation coefficient between TE1 and TE2 is as high as 0.8861, implying that the more efficient the production of deposits is, the more efficient is the second production stage for final products. Bank managers are recommended to increase their managerial ability at the first stage, optimizing the allocation of inputs, since this may foster cost efficiency in the second stage.
We can calculate the measure of output elasticity by summing up the partial derivatives of (log)output with respect to each (log)input, i.e.,
2
is greater than, equal to, or less than unity, then the technology exhibits increasing,‧
regarding the mean value of scale economies is equal to 1.93, reflecting that an average U.S. bank exhibits IRS in the first stage. We may conclude that the sample banks should keep expanding their production scale in order to enjoy the advantage of economies of scale, since doubling all their inputs would raise outputs by more than double.The coefficient estimates of the cost frontier in the second stage permit us to evaluate the measures of scale economies (SE) and scope economies (SC).11,12 The average SE measure is equal to 1.05 with a standard deviation of 0.0408. The U.S.
banks are, on average, operating under increasing returns to scale technology, which is consistent with the findings of Wheelock and Wilson (2012), Hughes and Mester (2013), and Restrepo et al. (2013), to mention a few. The sample banks benefit from economies of size and therefore are suggested to expand their production scale in order to reduce their long-run average cost. The average SC measure is equal to 0.07, indicating the presence of scope economies. It is preferable for those U.S. banks producing multiple outputs to concentrate on a single output or a few outputs. This finding tends to support the formation of financial conglomerates that are capable of providing an array of financial products within an organization in such a way as to share various resources, like computer equipments and clients’ information.
The application of the network DEA model requires knowing the distribution of
11 The formula of scale economies is written as
3 increasing, constant, or decreasing, when the SE is greater than, equal to, or less than unity.
12 Following Kim (1986), we formulate scope economies as:
(
12 ,
1 2,
2) ( ,
1 22 ,
2 3) ( ,
1 2,
32 )
3( , , ) / ( , , )
1 2 3
1 2 3SC C Y C Y C Y C Y Y Y C Y Y Y
, where
j(j =1, 2, 3) denotes 10% of the mean value of the jth output. If SC is greater than (less than) zero, then the economies (diseconomies) of the product mix prevail.‧ 國
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some inputs among alternative production stages - that is, the fractional parameters must be known, a priori. This requirement does not generally hold and cannot be estimated in the context of DEA. Holod and Lewis (2011) thus propose a modified model that avoids this requirement and can assess the efficiency of each bank at the expense of failing to evaluate the efficiency of each stage. Kao and Hwang (2010) impose restrictions on the values of fractional parameters in the range of [0.6, 0.9].
These restrictions may not be consistent with the true condition and hence give rise to undesirable estimation results. To validate this assertion, we re-estimate our model under the assumption that the values of
1 and
2 are arbitrarily given for several combinations of them. Table 6 shows the results of average TE scores in the two stages, which are denoted byTE and
1*TE , respectively, for different sets of
2*
1 and
2. The column “Diff” represents the difference in the average TE scores between (TE ,
1TE ) and (
2TE ,
1*TE ). For the case of
*2
1
0.5 and
2
0.5, i.e., half the labor and physical capital are consumed in the first stage, the average values ofTE and
1*TE
*2 are roughly 74% and 80%, respectively, which are significantly different from the corresponding average values of TE1 and TE2. The results reveal that the fractional parameters play important roles in determining efficiency scores in different stages. We also consider other cases, such as (
1= 0.26,
2= 0.26), (
1= 0.68,
2= 0.68), and (
1= 0.68,
2= 0.26), and the results are presented in the second to fourth rows ofTable A.6. All of the paired differences between (
TE
1,TE
2) and (TE ,
1*TE ) attain
2* statistical significance at the 1% level.We conduct another experiment, where the intersectoral production processes are assumed to be mutually independent, i.e., setting
= 0. The conclusion of the‧ 國
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statistical tests is still the same as above, as shown in the fifth row of Table 1.6. We finally consider the traditional treatment, i.e., banks employ a bundle of inputs to produce an array of outputs in a single stage. Efficiency scores can be evaluated either by a production function that contains two inputs, labor and capital, and an output, deposits, or by a cost function that includes three inputs, labor, capital, and funds, and three outputs, loans, investments, and non-interest income. Average efficiency measures are reported in the bottom two rows of Table 1.6. These average values are significantly different from those of
TE
1 andTE
2.Table 1.6 The mean TE scores of various sets of
1 and
2*
TE
1 Diff. *TE
2 Diff.
1= 0.50,
2= 0.50 0.74 -0.0649*** 0.80 -0.0267***
1= 0.26,
2= 0.26 0.70 -0.0315*** 0.90 -0.1257***
1= 0.68,
2= 0.68 0.77 -0.1014*** 0.93 -0.1511***
1= 0.68,
2= 0.26 0.75 -0.0757*** 0.95 -0.1741***
= 0 0.84 -0.1636*** 0.85 -0.0784***Production Function 0.88 -0.2059*** － －
Cost Function － － 0.93 -0.1546***
Note: *** denotes significance at the 1% level.
Figure 1.1 plots the kernel densities of the estimated TE scores for all cases considered, where Panel A corresponds to our empirical results with estimated
ˆ
1=0.26 and
ˆ
2= 0.68. The remaining panels of B to G draw the kernel density for each of the cases considered in Table 1.6, in which the fractional parameters are not estimated, but given. Obviously, shapes of the kernel densities of the remaining panels deviate substantially away from those in Panel A, implying that the distributions of those TE scores are indeed dissimilar.‧
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Figure 1.1 Kernel densities of estimated TE scores
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In sum, Table 1.6 and Figure 1.1 present that the mean TE is sensitive to different values of
1 and
2, depends on the presence of
, and varies with the assumptionof a single stage or multiple production stages. If