According to Lerner (1934), the Lerner Index (LI) assesses pricing ability by measuring the mark-up of output price over marginal cost. Its formula is defined as:
𝐿𝐼𝑖𝑡 = (𝑃𝑖𝑡− 𝑀𝐶𝑖𝑡)/𝑃𝑖𝑡⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (1) where 𝑃𝑖𝑡 denotes the output price of bank i at time t, and 𝑀𝐶𝑖𝑡 denotes the
marginal cost of its output, which is derived by taking the partial derivative of the cost function with respect to the single output. This requires estimating the translog cost function, specified as a function of a single output (Q) and three inputs (𝑊𝑘, 𝑘 = 1,2,3): of the half-normal random variable u, serving as the measure of cost inefficiency.
Variables α, β, η, γ, ω, φ, 𝜎𝑣12 , 𝑎𝑛𝑑 𝜎𝑢12 are unknown corresponding parameters to be estimated. As required by the microeconomic theory, restrictions such as symmetry and homogeneity of degree one in input prices are imposed to (2) prior to estimation.
The implied marginal cost function can be easily deduced by the following formula:
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𝑀𝐶𝑖𝑡 =𝑇𝐶𝑖𝑡
𝑄𝑖𝑡 �𝛼1+ 𝛼2ln 𝑄𝑖𝑡+ � 𝛾𝑘ln 𝑊𝑘𝑖𝑡 3
𝑘=1
+ 𝜔3𝑇𝑟𝑒𝑛𝑑� ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (3)
As mentioned in previous sections, the conventional approach that derives P and MC in formula (1) from separate channels is subject to the influence of random shocks, which frequently results in negative measure of the LI. Unfortunately, the negative value of the LI reveals that the firm sets price below its marginal cost, which appears to be implausible. To disentangle this problem, Huang et al. (2013) propose the copula-based econometric approach to quantify market power. They suggest constructing a price frontier to promise a non-negative Lerner Index measure.
As a profit-maximizing firm, whether or not in an imperfect competition market, an output price would be set no less than its MC, with the equality sign held in a perfect competition market. This implies that the following inequality must be satisfied:
P ≥ MR = MC ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (4) After appending a non-negative random variable 𝑢2~|𝑁(0, 𝜎𝑢22 )| and a statistical noise term 𝑣2~𝑁(0, 𝜎𝑣22 ) to the right-hand side of (4), we obtain:
𝑃𝑖𝑡 = 𝑀𝐶𝑖𝑡+ 𝑣2𝑖𝑡 + 𝑢2𝑖𝑡⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (5) The composed error term 𝜀2 = 𝑣2 + 𝑢2 is allowed to be correlated with 𝜀1 in (2).
Note that now 𝑀𝐶 + 𝑣2 forms the stochastic price frontier which is similar in format to the standard production or cost frontier. In this manner, the one-sided error 𝑢2 reflects the deviation of the output price from the MC. This gap can be estimated by the conditional expectation of E(𝑢2𝑖𝑡|𝜀2𝑖𝑡).14 The larger the conditional expectation, the stronger the market power of a firm has and the larger the abnormal profit is
14 The conditional expectation can be shown to be equal to E(𝑢2𝑖𝑡|𝜀2𝑖𝑡) = µ2∗𝑖𝑡+ 𝜎2∗ ∅(−𝑢2∗𝑖𝑡⁄𝜎2∗)
1−𝜙(−𝑢2∗𝑖𝑡⁄𝜎2∗)
where 𝜎2∗2= 𝜎𝑢22 𝜎𝑣22⁄ , 𝜎𝜎22 22= 𝜎𝑢22 + 𝜎𝑣22, µ2∗𝑖𝑡= − 𝜎𝑢22 𝜀2𝑖𝑡⁄ . 𝜎22
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expected to be earned.
Formula (2) and (5) are recommended to be estimated simultaneously to avoid potential bias and to improve the efficiency of parameter estimates, as mentioned by Lai and Huang (2010). Their system regression model is also referred to as the seemingly unrelated stochastic frontier model and allows for dependent composite errors. The LI is then computed by E(𝑢2𝑖𝑡|𝜀2𝑖𝑡)/𝑃𝑖𝑡. To derive the joint pdf and the corresponding log-likelihood function for the dependent composite errors, Huang et al.
(2013) applied the copula method, as proposed by Lai and Huang (2010), which requires using the approximation approach suggested by Tsay et al. (2012). Please refer to the Appendix 1 for the detailed derivation of the joint pdf and log-likelihood function.
3.2 The Nexus of NIM and Noninterest Income
Laeven and Levin (2007), Baele et al. (2007), and Nguyen (2012) have
addressed the endogeneity problem between incomes of traditional and non-traditional activities. Yet prior works address this problem in the context of a single equation. To deal with the endogeneity issue under the framework of the simultaneous equations, which consists of two equations of the NIM and noninterest income, we apply the generalized method of moments (GMM) to estimate the model, as proposed by Lewbel (2012). While tackling with simultaneous equations, the identification of structural coefficients is pivotal. Once the identification is achieved, consistent estimators can be obtained and various statistical inferences of interest can be performed. Thanks to Lewbel (2012), we are capable of obtaining identification by restricting correlations of εε′ with exogenous regressors X. A salient feature of Lewbel’s method is that it does not rely on the rank or order conditions to identify structural parameters. Rather, his approach identifies the parameters of interest by
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assuming heteroskedastic error terms. His model is specified as:
𝑌1 = 𝑋′𝛽1+ 𝑌2𝛾1+ 𝜀1, 𝜀1 = 𝛼1𝑈 + 𝑉1⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (19) 𝑌2 = 𝑋′𝛽2+ 𝑌1𝛾2+ 𝜀2, 𝜀2 = 𝛼2𝑈 + 𝑉2⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (20) where 𝑈, 𝑉1, and 𝑉2 are unobserved variables uncorrelated with exogenous
regressors X and are conditionally uncorrelated with each other. 𝑉1 and 𝑉2 are idiosyncratic errors in the two equations. The correlation between 𝜀1 and 𝜀2 is demonstrated via 𝑈, which is an omitted variable or other unobserved factor that may directly influence both 𝑌1 and 𝑌2. Let Z be a vector of observed exogenous variables, which could be a sub-vector of X, or could equal X. The following assumptions are imposed: (1) X is uncorrelated with (𝑈, 𝑉1, 𝑉2); (2) Z is uncorrelated
with �𝑈2, 𝑈𝑉𝑗, 𝑉1𝑉2�; (3) Z is correlated with 𝑉12 and 𝑉22. Given these assumptions, the following conditions are satisfied:
Cov(𝑍, 𝜀1𝜀2) = 𝐶𝑜𝑣(𝑍, 𝛼1𝛼2𝑈2+ 𝛼1𝑈𝑉2+ 𝛼2𝑈𝑉1+ 𝑉1𝑉2) = 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (21) Cov(𝑍, 𝜀22) = 𝐶𝑜𝑣�𝑍, 𝛼22𝑈2+ 2𝛼2𝑈𝑉2+ 𝑉22� = 𝐶𝑜𝑣(𝑍, 𝑉22) ≠ 0 ⋯ ⋯ ⋯ ⋯ ⋯ (22) Equations (21) and (22) are exactly the requirements for applying Lewbel (2012)’s identification theorems and associated estimators. Only moments and some
heteroskedasticity of 𝜀𝑗 (j =1, 2) are required for identification and estimation, i.e., E(𝑋𝜀1) = 0, 𝐸(𝑋𝜀2) = 0, 𝑐𝑜𝑣(𝑍, 𝜀1𝜀2) = 0 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (23) The moments provide identification whether or not Z is a sub-vector of X.
Let µ = E(𝑍), 𝜃 = (𝛾1, 𝛾2, 𝛽1, 𝛽2, 𝜇)′, and S includes all the elements of Y, X, and Z. Define the function of vectors as:
𝑓1(𝑆, 𝜃) = 𝑋(𝑌1− 𝑋′𝛽1− 𝑌2𝛾1) 𝑓2(𝑆, 𝜃) = 𝑋(𝑌2− 𝑋′𝛽2 − 𝑌1𝛾2)
𝑓3(𝑆, 𝜃) = 𝑍 − 𝜇
𝑓4(𝑆, 𝜃) = (𝑍 − 𝜇)(𝑌1− 𝑋′𝛽1− 𝑌2𝛾1)(𝑌2− 𝑋′𝛽2− 𝑌1𝛾2)
Stack all the four functions of vectors to one single function of vectors 𝑓(𝑆, 𝜃). Let Θ
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be the set of all 𝜃, where 𝜃0 ∈ Θ is the true mean of 𝜃. When 𝜃 = 𝜃0 the moment condition E[𝑓(𝑆, 𝜃)] = 0 is satisfied. If there are T observations of random
sample 𝑆1, ⋯ , 𝑆𝑇, we can define the sample moments as: 𝑓𝑇(𝜃) = 𝑇−1∑𝑇𝑡=1𝑓(𝑆𝑡, 𝜃). The GMM estimators of 𝜃 are expressed as 𝜃� = argmin𝑇 𝜃𝑄𝑇(𝜃), where 𝑄𝑇(𝜃) = 𝑓𝑇(𝜃)′𝑉� 𝑓𝑇−1 𝑇(𝜃). If and only if 𝑉� is a positive definite matrix and 𝑉𝑇−1 �(𝜃) =𝑇 1
𝑇∑𝑇𝑡=1𝑓(𝑆𝑡, 𝜃)𝑓(𝑆𝑡, 𝜃)′, 𝜃� are consistent and asymptotically efficient estimators of 𝑇
𝜃 with an asymptotic covariance matrix (𝐹′𝑉−1𝐹)−1, where V = Var[𝑓(𝑆𝑡, 𝜃0)] and F = E �𝜕𝑓(𝑆,𝜃)𝜕𝜃′ � is the non-random matrix.
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