In the case of the cost function where input prices and outputs are taken as predetermined or in the case of (
4) where the covariates are weakly exogenous, the Bayesian techniques, we have described, can be applied easily. However,
there are many instances in which the covariates or explanatory variables are endogenously determined. An extremely important case is when (
4) represents a production function. Under the assumption of cost minimization, inputs are decision variables (and, therefore, endogenous), while output (the left-hand-side variable) is predetermined. Moreover, economic exogeneity and econometric exogeneity are different things. If econometric exogeneity is rejected, this does not mean that the economic assumptions are wrong. Measurement errors, for example, would be a typical example. Lai and Kumbhakar (
2019) consider the case of a Cobb–Douglas production function along with the first-order conditions for cost minimization. To summarize the approach of Lai and Kumbhakar (
2019), we have the following Cobb–Douglas production function:
$$\begin{aligned} y_{it}=\beta _{0}+\sum _{k=1}^{K}\beta _{k}x_{kit}+v_{it}-u_{it},i=1,\ldots ,n,t=1,\ldots ,T, \end{aligned}$$
(32)
where
\(y_{it}\) is log output for firm
i and date
t,
\(x_{kit}\) is the log of
kth input for firm
i and date
t ,
\(v_{it}\) is a two-sided error term,
\(u_{it}\) is a non-negative error component that represents technical inefficiency in production, and
\(\beta _{k}>0,k=1,\ldots ,K\). Suppose also input prices are
\(w_{kit}.\) From the first-order conditions of cost minimization (where inputs are endogenous choice variables and output is predetermined), we obtain:
$$\begin{aligned} \frac{\partial y_{it}/\partial x_{kit}}{\partial y_{it}/\partial x_{1it}}=\frac{w_{kit}x_{kit}}{w_{1it}x_{1it}}=\frac{\beta _{k}}{\beta _{1}}\mathrm{e}^{v_{kit}},2=1,\ldots ,K. \end{aligned}$$
(33)
These conditions can be expressed as follows:
$$\begin{aligned} x_{1it}-x_{kit}=\ln (w_{kit}/w_{1it})+(\ln \beta _{1}-\ln \beta _{j})+v_{kit},k=2,\ldots ,K. \end{aligned}$$
(34)
The constraints are only on the parameters
\(\beta _{k}\) (
\(k=1,\ldots ,K\)), so this is not a very interesting example. However, if we generalize (
32) to the translog case, we have:
$$\begin{aligned}&y_{it}=\beta _{0}+\sum _{k=1}^{K}\beta _{k}x_{kit}+\tfrac{1}{2}\nonumber \\&\sum _{k=1}^{K}\sum _{k'=1}^{K}\beta _{kk'}x_{kit}x_{k'it}+v_{it}-u_{it},i=1,\ldots ,n,t=1,\ldots ,T. \end{aligned}$$
(35)
Suppose all parameters are collected into the vector
\(\beta \) whose dimension is
\(d=1+K+\tfrac{K(K+1)}{2}\), after imposing symmetry, viz.
\(\beta _{kk'}=\beta _{k'k},\,k,k'=1,\ldots ,K\). The first-order conditions for cost minimization are as follows:
$$\begin{aligned} \frac{\partial y_{it}/\partial x_{kit}}{\partial y_{it}/\partial x_{1it}}=\frac{w_{kit}x_{kit}}{w_{1it}x_{1it}}=\frac{\beta _{k}+\sum _{k'=1}^{K}\beta _{kk'}\ln x_{k',it}}{\beta _{1}+\sum _{k'=1}^{K}\beta _{1k'}\ln x_{k',it}}\mathrm{e}^{v_{kit}},k=2,\ldots ,K. \end{aligned}$$
(36)
These equations can be rewritten as follows:
$$\begin{aligned}&x_{kit}-x_{1it}= \ln \left( \beta _{k}+\sum \limits _{k'=1}^{K}\beta _{kk'}x_{k',it}\right) \nonumber \\&\quad -\ln \left( \beta _{1}+\sum \limits _{k'=1}^{K}\beta _{1k'}x_{k',it}\right) -\ln (w_{kit}/w_{1it})+v_{kit},\,k=2,\ldots ,K. \end{aligned}$$
(37)
Moreover, it is convenient to rewrite (
35) in the form:
$$\begin{aligned} y_{it}=\psi (x_{it})\beta +v_{it}-u_{it},\,i=1,\ldots ,n,t=1,\ldots ,T, \end{aligned}$$
(38)
where
\(\psi (x_{it})=[1,x_{1it},\ldots ,x_{Kit},\tfrac{1}{2}x_{1it}^{2},\ldots ,\tfrac{1}{2}x_{Kit}^{2},x_{1it}x_{2it},\ldots ,x_{(K-1)it}x_{Kit}]'\) are the nonlinear terms in the translog functional form.
The economic restrictions are as follows. First, we have monotonicity:
$$\begin{aligned} \beta _{k}+\sum _{k'=1}^{K}\beta _{kk'}x_{k',it}\ge 0,\,k=1,\ldots ,K. \end{aligned}$$
(39)
This imposes the following set of restrictions:
$$\begin{aligned} 0=\beta _{k}+\sum _{k'=1}^{K}\beta _{kk'}x_{k',it}+{\tilde{v}}_{k,it}+{\tilde{u}}_{k,it},\,k=1,\ldots ,K. \end{aligned}$$
(40)
Following Diewert and Wales (
1987), given the monotonicity restrictions, we need the matrix
\(B=[\beta _{kk'},k,k'=1,\ldots ,K]\) to be negative semi-definite. Therefore, it is necessary and sufficient that the eigenvalues of
B, say
\(\varLambda (\beta )=[\lambda _{1}(\beta ),\ldots ,\lambda _{K}(\beta )]'\), are all non-positive. This imposes the following set of nonlinear restrictions:
$$\begin{aligned} -\mathrm {vec}\varvec{\varLambda }(\beta )_{(TK\times 1)}=v_{(TK\times 1)}^{o}+u_{(TK\times 1)}^{o}. \end{aligned}$$
(41)
From (
40) and (
41), we have 2
K additional equations so in total we have
K endogenous variables but 3
K stochastic equations. From the econometric point of view, this is, clearly,
a major problem,
as we lack 2
K endogenous variables to complete the system in (
35), (
37), (
40),
and (
41). Let us write the entire system as follows:
$$\begin{aligned} \left[ \begin{array}{c} -y_{it}\\ x_{2it}-x_{1it}\\ \vdots \\ x_{Kit}-x_{1it}\\ {\mathbf {0}}_{K}\\ {\mathbf {0}}_{K} \end{array}\right] =\left[ \begin{array}{c} -\psi (x{}_{it})\beta \\ g_{2}(x_{it};\beta )\\ \vdots \\ g_{K}(x_{it};\beta )\\ {\mathbf {m}}(x_{it};\beta )\\ {\mathbf {s}}(x_{it};\beta ) \end{array}\right] +\left[ \begin{array}{c} v_{it,1}\\ v_{it,2}\\ \vdots \\ v_{it,K}\\ {\tilde{v}}_{it,(K\times 1)}\\ v_{it,(K\times 1)}^{o} \end{array}\right] +\left[ \begin{array}{c} u_{it,1}\\ u_{it,2}\\ \vdots \\ u_{it,K}\\ {\tilde{u}}_{it,(K\times 1)}\\ {\check{u}}_{it,(K\times 1)} \end{array}\right] \end{aligned}$$
(42)
where
\(g_{k}(x_{it};\beta )=\ln \left( \beta _{k}+\sum _{k'=1}^{K}\beta _{kk'}\ln x_{k',it}\right) -\ln \left( \beta _{1}+\sum _{k'=1}^{K}\beta _{1k'}\ln x_{k',it}\right) -\ln (w_{kit}/w_{1it}),\,k=2,\ldots ,K\),
\({\mathbf {m}}(x_{it};\beta )=[m_{1}(x_{it};\beta ),\ldots ,m_{K}(x_{it};\beta )]'\),
\(m_{k}(x_{it};\beta )=\beta _{k}+\sum _{k'=1}^{K}\beta _{kk'}\ln x_{k',it},\,k=1,\ldots ,K\),
\({\mathbf {s}}(x_{it};\beta )=[s_{1}(x_{it};\beta ),\ldots ,s_{K}(x_{it};\beta )]'\),
\(s_{k}(x_{it};\beta )=\lambda _{k}(x_{it};\beta ),\,k=1,\ldots ,K\). Let us write the system in (
42) compactly as follows:
$$\begin{aligned} {\mathbf {Y}}_{it}={\mathbf {f}}(x_{it};\beta )+{\mathbf {v}}_{it}+{\mathbf {u}}_{it}. \end{aligned}$$
(43)
Although we have 3
K equations, there are only
K endogenous variables. To provide 2
K additional equations, it seems that the only possibility is to assume that
\({\mathbf {U}}_{it}\equiv [{\tilde{u}}_{it}',{\check{u}}'_{it}]'\) are, in fact,
endogenous variables.
This provides, indeed, the missing 2K additional equations. To accomplish this, we depart from the assumption that
\({\mathbf {U}}_{it}\) is a (vector) random variable, and, instead, we make use of the following device originally proposed by Paul and Shankar (
2018) and further developed by Tsionas and Mamatzakis (
2019):
$$\begin{aligned} {\mathbf {U}}_{it}=\left[ \begin{array}{c} U_{it,1}\\ \vdots \\ U_{it,2K} \end{array}\right] =\left[ \begin{array}{c} -\ln \varPhi (x'_{it}\gamma _{1})\\ \vdots \\ -\ln \varPhi (x'_{it}\gamma _{K}) \end{array}\right] \equiv {\mathbf {h}}(x_{it};\gamma ), \end{aligned}$$
(44)
where
\(\varPhi (\cdot )\) is any distribution function (for example, the standard normal), and
\(\gamma =[\gamma '_{1},\ldots ,\gamma _{K}']'\in {\mathbb {R}}^{2K}\) is a vector of parameters. The idea of Paul and Shankar (
2018) is that efficiency,
\(r=\mathrm{e}^{-u}\) is in the interval (0, 1] and, therefore,
r can be modeled using any distribution function. In turn, we have:
$$\begin{aligned} \left[ \begin{array}{c} -y_{it}\\ \Delta x_{it}\\ {\widetilde{u}}_{it} \end{array}\right] =\left[ \begin{array}{c} -\psi (x_{it})\beta \\ {\mathbf {g}}(x_{it};\beta )\\ {\mathbf {h}}(x_{it};\beta ,\gamma ) \end{array}\right] +\left[ \begin{array}{c} v_{it,1}\\ {\widetilde{v}}_{it}\\ \mathring{v}_{it,(2K\times 1)} \end{array}\right] +\left[ \begin{array}{c} u_{it,1}\\ {\widetilde{u}}_{it}\\ {\mathbf {0}}_{(2K\times 1)} \end{array}\right] , \end{aligned}$$
(45)
where
\(\mathring{v}_{it}=[{\tilde{v}}'_{it},{\check{v}}'_{it}]'\),
\({\mathbf {V}}_{it}=[v_{it,1},\ldots ,v_{it,K},\mathring{v}_{it}']'\),
\({\mathbf {U}}_{it}=[u_{it,1},\ldots ,u_{it,K}]'\),
\(\triangle x_{it}=[x_{2it}-x_{1it},\ldots ,x_{Kit}-x_{1it}]'\),
\({\mathbf {g}}(x_{it};\beta )=[g_{2}(x_{it};\beta ),\ldots ,g_{K}(x_{it};\beta )]'\),
\({\widetilde{v}}_{it}=[v_{it,2},\ldots ,v_{it,K}]'\),
\({\widetilde{u}}_{it}=[u_{it,2},\ldots ,u_{it,K}]'\). MCMC for this model is detailed in the “Technical Appendix” (section 5.4).