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Abstract
This article studies a partially linear seemingly unrelated regressions (SUR) model to estimate a translog cost system that consists of a partially linear translog cost function and input share equations. The parametric component is estimated via a simple two-step feasible SUR estimation procedure. We show that the resulting estimator achieves root-n convergence and is asymptotically normal. The nonparametric component is estimated with a nonparametric SUR estimator based on the Cholesky decomposition. We show that this estimator is consistent, asymptotically normal, and more efficient relative to the ones that ignore cross-equation correlation. We emphasize the importance and implication of the choice of square root of the covariance matrix by comparing the Cholesky and Spectral decompositions. A model specification test for parametric functional form is proposed. An Italian banking data set is used to estimate the translog cost system. Results show that marginal effects of risks on cost of production are heterogeneous but increase with risk levels.
Acknowledgements
This article was presented at the University of Colorado Boulder, 2021 Asian Meeting of the Econometric Society, and 4th International Conference on Econometrics and Statistics. We are grateful to Christos Ioannidis, Subal C. Kumbhakar, Carlos Martins-Filho, and four anonymous referees for their helpful comments.
Notes
1 The translog parameters have few economic meanings because they are not elasticities on their own. Therefore, it would be costly to make these parameters unknown functions of zi as in Henderson et al. (Citation2015). The input price and output elasticities derived from the translog parameters are more economically meaningful, and are observation-specific so long as the cost function is translog in input prices and outputs, even if the slope translog coefficients are constants rather than nonparametric functions.
2 An anonymous referee correctly points out that a prerequisite of achieving efficiency improvement by estimating a system of equations is that each equation in the system is correctly specified which leads to consistent pilot estimates. Therefore, it is recommended that economic theory be utilized in motivating the specification of each equation in the system or the relationships between the equations in the system.
3 The cost function can be derived from a standard cost minimization problem, i.e., subject to a transformation function,
The first-order conditions would imply that
and
Let
then
4 Note that due to data requirements this article considers share equations with respect to input prices only. It would be possible to consider an extended setting where, subject to data availability, the nonparametric component of one equation in a system is the first-order partial derivative of a nonparametric component in another equation in the system. We leave this for future research.
5 The last share equation, SJ, is dropped because the sum of the cost shares equals unity. If the zi in Equation(2.6)(2.6)
(2.6) is viewed as fixed, then the partially linear SUR is equivalent to a parametric SUR model. The equation-by-equation OLS estimator that we use in the first step is invariant to the choice of the numeraire, and also to the equation dropped (Chavas and Segerson, Citation1987). Therefore, the residuals from the first-step estimation, and also the estimated variance–covariance matrix, would be invariant to the choice of the numeraire. The case of zi being viewed as stochastic and its impact on the choice of numeraire is saved for future research.
6 The translog cost system can be used to represent technology for an industry whose outputs are most likely to be exogenous (e.g., banks, airlines, railroads, post offices, and public utilities) rather than endogenous (e.g., manufacturing firms, agricultural farms without explicit quotas on outputs) (Kumbhakar, Citation2013). The moment conditions stated in Section 3 must be satisfied to estimate the translog cost system—an illustrative example of the partially linear SUR model—with the proposed estimators.
7 For example, if we have three inputs and three outputs, we can write the parameter vector of the translog cost function, Equation(2.6)(2.6)
(2.6) , as
for the variable vector of the cost function, i.e.,
then the right-hand-side variable vectors of the share equations, Equation(2.7)
(2.7)
(2.7) , can be written as:
for j = 1, and
for j = 2. The R codes of estimating the partially linear SUR model are available from the authors upon request.
8 For example, if we have three inputs and one output, we can write the parameter vector of the translog profit function, Equation(2.13)(2.13)
(2.13) , as
for the variable vector of the profit function, i.e.,
then the right-hand-side variable vectors of the share equations, Equation(2.14)
(2.14)
(2.14) , can be written as:
for j = 1,
for j = 2, and
for j = 3.
9 The asymptotic distribution of can be obtained easily from the joint one of
It is normal with asymptotic covariance being the corresponding
diagonal block of
10 We need to undersmooth in the first step so that the nonparametric bias term becomes negligible when it comes to estimating the parametric coefficients β.
11 We thank an anonymous referee for pointing out these relevant articles to better position our article in the literature.
12 It is well known that the efficiency of the GLS estimator for fully linear models is invariant to the covariance decomposition method since the estimator only depends on the inverse of the covariance matrix rather than on its square root; see, e.g., EquationEq. (3.5)(3.5)
(3.5) . More interestingly, we find that how to decompose the error covariance matrix might play a role in the extent to which efficiency of the nonparametric estimation improves—the resultant nonparametric SUR estimator explicitly depends on the inverse of the square root of the covariance matrix; see Remark 6.
13 This originates from a spherical transformation of EquationEq. (3.7)(3.7)
(3.7) where
and H is a diagonal matrix with the same diagonal elements as V; see Su et al. (Citation2013) for more details.
14 It can be easily shown that a local-polynomial kernel estimation of equation can be achieved by treating
as the regressand.
15 Intuitively, the Cholesky decomposition yields a lower triangular Cholesky factor, which allocates most (least) information to the transformed dependent variable in the last (first) equation of a system, while the Spectral decomposition yields a symmetric Spectral factor, which allocates information equally to each transformed dependent variable in the system. To put it another way, Cholesky allocates all the covariance information to the last equation by not providing any covariance information to the first equation, while Spectral gives some covariance information to each and every equation.
16 We would like to thank an anonymous referee for the suggestion.
17 The proofs of these results are similar to those in Li and Wang (Citation1998), and therefore omitted.
18 This bootstrap procedure also works for the case where zsi—which are elements of zi—are different across equations. For the translog cost system, only the cost function has a nonparametric component in it, i.e., the nonparametric components in the share equations are restricted to be zeros. In this case, the test statistic would be based on the single equation of cost function, as shown in Remark 9, and therefore, only the information from the cost function would be required to compute the test statistic, and the bootstrap procedure would be exactly the same as that in Li and Wang (Citation1998).
19 To guarantee that the bootstrap errors satisfy the regression conditions, first re-center the residuals from the null model and obtain where
then use the wild bootstrap method (Liu, Citation1988; Mammen, Citation1993) such that
with probability
and
with probability
As an aside, note that heteroscedasticity is embedded in a SUR system in general given the Σm with different elements. Also for a system of equations, we can view each i as a cluster and the same wild bootstrap weight is applied to each cluster across the m equations (Cameron and Trivedi, Citation2005, Chapter 11).
20 We also consider the no cross-equation correlation case, i.e., and find that for all sample sizes, the parametric and nonparametric component estimators have similar MSEs and AMSEs, respectively, and therefore results are omitted to save space, but are available upon request.
21 In fact, for the Cholesky decomposition, the first equation’s asymptotic efficiency is not improved at all—it has the same asymptotic variance as the single-equation counterpart.
22 Precisely, this alternative equation system is for which the estimated error covariance matrix is
23 See for details.
24 Purchased funds such as customer deposits are viewed as inputs which are used to produce loans and other assets. See Hughes and Mester (Citation1993) for more details about treating deposits as inputs in estimating a cost function.
25 Risk in banking is quantified by the amount of financial capital reserved by a bank. A higher level of financial capital means that a bank is exposed to a lower level of risk and better protects a bank from failure. The financial capital includes (1) loan loss provisions that are used to cover loan losses from, say, non-performing loans, (2) equity capital paid by investors, e.g., shareholders, for a bank to meet its long-term debt obligations, and (3) liquid assets that can easily be converted into cash in a short amount of time for a bank to repay its debt in short notice. Bank managers’ willingness to take risk would determine the level of risk, i.e., the level of financial capital that a bank reserves, and therefore risk in banking is observable and controlled by bank managers. It is widely accepted in the banking literature that banking risk is listed as one of the arguments of the cost function (Mester, Citation1996). In this article, banking risks are viewed as environmental variables that affect total cost in flexible manners. This is because the bank managers’ risk preferences would determine how they respond to risk, and hence the banks’ production environment. These underlying heterogeneous risk preferences would generate different marginal effects of risk on cost.
26 See Resti (Citation1997) for a brief introduction to the Italian banking system.
27 The underlying cost function with non-neutral effects in its original unrestricted form is
28 Given that the cost share of the numeraire input is the non-neutral effects of the
risk on SJi is
In addition, the marginal costs are calculated as
where captures the non-neutral effects of the
risk on the marginal cost of producing the
output,
and
See Appendix E for alternative cost functions with non-neutral effects of time on the cost shares of inputs and marginal costs of outputs.
29 A caution is that too many share equations might inflate the estimated error variance in practice.
30 The figures under the alternative cost function in Equation(5.1)(5.1)
(5.1) are omitted to save space, but are available upon request.
31 For the translog cost system, the nonparametric component is in the cost function only. The cost function is placed as the last equation of the cost system. Therefore, in this particular example with two share equations,
32 See also Theorem 2 of Geng et al. (Citation2020).
33 We also consider the no cross-equation correlation case, i.e., and find that for all sample sizes, the parametric and nonparametric component estimators have similar MSEs and AMSEs, respectively, and therefore results are omitted to save space, but are available upon request.
34 Nonlinear effects of risks (time) on cost shares and marginal costs can be generated by interacting higher order risk variables (time) with input prices and outputs, respectively.